Shared Qs (021)

1. 
   

   Question

   prevnext

   Does the complex number \(z=3+6i\) solve the equation below?

   \[z^2-6z+47=0\]

   
   
   1. Yes
   2. No
   
   

   Solution

   Show solution

   Plug in \(z=3+6i\). Since we are checking if this equation is true, we should probably put a question mark above the equal sign. I didn’t, but keep in mind these equations might be wrong.

   \[(3+6i)^2-6(3+6i)+47=0\]

   Expand the terms. Use FOIL for the squared term.

   \[9+18i+18i+36i^2-18-36i+47=0\]

   Remember \(i^2=-1\). \[9+18i+18i-36-18-36i+47=0\] Combine like terms. \[2+0i=0\] Nope! The complex number does not satisfy the equation!

   
   
2. 
   

   Question

   prevnext

   Does the complex number \(z=5-3i\) solve the equation below?

   \[z^2-10z+34=0\]

   
   
   1. Yes
   2. No
   
   

   Solution

   Show solution

   Plug in \(z=5-3i\). Since we are checking if this equation is true, we should probably put a question mark above the equal sign. I didn’t, but keep in mind these equations might be wrong.

   \[(5-3i)^2-10(5-3i)+34=0\]

   Expand the terms. Use FOIL for the squared term.

   \[25-15i-15i+9i^2-50+30i+34=0\]

   Remember \(i^2=-1\). \[25-15i-15i-9-50+30i+34=0\] Combine like terms. \[0+0i=0\] Yay! The complex number satisfies the equation!

   
   
3. 
   

   Question

   prevnext

   Does the complex number \(z=-2-8i\) solve the equation below?

   \[z^2+3z+72=0\]

   
   
   1. Yes
   2. No
   
   

   Solution

   Show solution

   Plug in \(z=-2-8i\). Since we are checking if this equation is true, we should probably put a question mark above the equal sign. I didn’t, but keep in mind these equations might be wrong.

   \[(-2-8i)^2+3(-2-8i)+72=0\]

   Expand the terms. Use FOIL for the squared term.

   \[4+16i+16i+64i^2-6-24i+72=0\]

   Remember \(i^2=-1\). \[4+16i+16i-64-6-24i+72=0\] Combine like terms. \[6+8i=0\] Nope! The complex number does not satisfy the equation!

   
   
4. 
   

   Question

   prevnext

   Does the complex number \(z=-7-9i\) solve the equation below?

   \[z^2+15z+127=0\]

   
   
   1. Yes
   2. No
   
   

   Solution

   Show solution

   Plug in \(z=-7-9i\). Since we are checking if this equation is true, we should probably put a question mark above the equal sign. I didn’t, but keep in mind these equations might be wrong.

   \[(-7-9i)^2+15(-7-9i)+127=0\]

   Expand the terms. Use FOIL for the squared term.

   \[49+63i+63i+81i^2-105-135i+127=0\]

   Remember \(i^2=-1\). \[49+63i+63i-81-105-135i+127=0\] Combine like terms. \[-10-9i=0\] Nope! The complex number does not satisfy the equation!

   
   
5. 
   

   Question

   prevnext

   Does the complex number \(z=-2-4i\) solve the equation below?

   \[z^2+4z+20=0\]

   
   
   1. Yes
   2. No
   
   

   Solution

   Show solution

   Plug in \(z=-2-4i\). Since we are checking if this equation is true, we should probably put a question mark above the equal sign. I didn’t, but keep in mind these equations might be wrong.

   \[(-2-4i)^2+4(-2-4i)+20=0\]

   Expand the terms. Use FOIL for the squared term.

   \[4+8i+8i+16i^2-8-16i+20=0\]

   Remember \(i^2=-1\). \[4+8i+8i-16-8-16i+20=0\] Combine like terms. \[0+0i=0\] Yay! The complex number satisfies the equation!

   
   
6. 
   

   Question

   prevnext

   Does the complex number \(z=-4-6i\) solve the equation below?

   \[z^2+9z+50=0\]

   
   
   1. Yes
   2. No
   
   

   Solution

   Show solution

   Plug in \(z=-4-6i\). Since we are checking if this equation is true, we should probably put a question mark above the equal sign. I didn’t, but keep in mind these equations might be wrong.

   \[(-4-6i)^2+9(-4-6i)+50=0\]

   Expand the terms. Use FOIL for the squared term.

   \[16+24i+24i+36i^2-36-54i+50=0\]

   Remember \(i^2=-1\). \[16+24i+24i-36-36-54i+50=0\] Combine like terms. \[-6-6i=0\] Nope! The complex number does not satisfy the equation!

   
   
7. 
   

   Question

   prevnext

   Does the complex number \(z=6-8i\) solve the equation below?

   \[z^2-12z+104=0\]

   
   
   1. Yes
   2. No
   
   

   Solution

   Show solution

   Plug in \(z=6-8i\). Since we are checking if this equation is true, we should probably put a question mark above the equal sign. I didn’t, but keep in mind these equations might be wrong.

   \[(6-8i)^2-12(6-8i)+104=0\]

   Expand the terms. Use FOIL for the squared term.

   \[36-48i-48i+64i^2-72+96i+104=0\]

   Remember \(i^2=-1\). \[36-48i-48i-64-72+96i+104=0\] Combine like terms. \[4+0i=0\] Nope! The complex number does not satisfy the equation!

   
   
8. 
   

   Question

   prevnext

   Does the complex number \(z=8-3i\) solve the equation below?

   \[z^2-16z+73=0\]

   
   
   1. Yes
   2. No
   
   

   Solution

   Show solution

   Plug in \(z=8-3i\). Since we are checking if this equation is true, we should probably put a question mark above the equal sign. I didn’t, but keep in mind these equations might be wrong.

   \[(8-3i)^2-16(8-3i)+73=0\]

   Expand the terms. Use FOIL for the squared term.

   \[64-24i-24i+9i^2-128+48i+73=0\]

   Remember \(i^2=-1\). \[64-24i-24i-9-128+48i+73=0\] Combine like terms. \[0+0i=0\] Yay! The complex number satisfies the equation!

   
   
9. 
   

   Question

   prevnext

   Does the complex number \(z=3+9i\) solve the equation below?

   \[z^2-6z+82=0\]

   
   
   1. Yes
   2. No
   
   

   Solution

   Show solution

   Plug in \(z=3+9i\). Since we are checking if this equation is true, we should probably put a question mark above the equal sign. I didn’t, but keep in mind these equations might be wrong.

   \[(3+9i)^2-6(3+9i)+82=0\]

   Expand the terms. Use FOIL for the squared term.

   \[9+27i+27i+81i^2-18-54i+82=0\]

   Remember \(i^2=-1\). \[9+27i+27i-81-18-54i+82=0\] Combine like terms. \[-8+0i=0\] Nope! The complex number does not satisfy the equation!

   
   
10. 
    

    Question

    prevnext

    Does the complex number \(z=-7-8i\) solve the equation below?

    \[z^2+14z+113=0\]

    
    
    1. Yes
    2. No
    
    

    Solution

    Show solution

    Plug in \(z=-7-8i\). Since we are checking if this equation is true, we should probably put a question mark above the equal sign. I didn’t, but keep in mind these equations might be wrong.

    \[(-7-8i)^2+14(-7-8i)+113=0\]

    Expand the terms. Use FOIL for the squared term.

    \[49+56i+56i+64i^2-98-112i+113=0\]

    Remember \(i^2=-1\). \[49+56i+56i-64-98-112i+113=0\] Combine like terms. \[0+0i=0\] Yay! The complex number satisfies the equation!

    
    
11. 
    

    Question

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    A student has taken 6 exams and gotten an average of 72. What score does the student need on the next exam to bring the average to 75? (All exams are equally weighted in the average.)

    
    

    Solution

    Show solution

    Find the current total. \[6 \cdot 72 = 432\]

    Find the necessary total. \[7\cdot75 = 525\]

    Find the difference of the totals. \[525-432 = 93\]

    The answer is 93.

    
    
12. 
    

    Question

    prevnext

    A student has taken 3 exams and gotten an average of 72. What score does the student need on the next exam to bring the average to 75? (All exams are equally weighted in the average.)

    
    

    Solution

    Show solution

    Find the current total. \[3 \cdot 72 = 216\]

    Find the necessary total. \[4\cdot75 = 300\]

    Find the difference of the totals. \[300-216 = 84\]

    The answer is 84.

    
    
13. 
    

    Question

    prevnext

    A student has taken 5 exams and gotten an average of 66.6. What score does the student need on the next exam to bring the average to 70? (All exams are equally weighted in the average.)

    
    

    Solution

    Show solution

    Find the current total. \[5 \cdot 66.6 = 333\]

    Find the necessary total. \[6\cdot70 = 420\]

    Find the difference of the totals. \[420-333 = 87\]

    The answer is 87.

    
    
14. 
    

    Question

    prevnext

    A student has taken 5 exams and gotten an average of 66. What score does the student need on the next exam to bring the average to 70? (All exams are equally weighted in the average.)

    
    

    Solution

    Show solution

    Find the current total. \[5 \cdot 66 = 330\]

    Find the necessary total. \[6\cdot70 = 420\]

    Find the difference of the totals. \[420-330 = 90\]

    The answer is 90.

    
    
15. 
    

    Question

    prevnext

    A student has taken 4 exams and gotten an average of 63.5. What score does the student need on the next exam to bring the average to 70? (All exams are equally weighted in the average.)

    
    

    Solution

    Show solution

    Find the current total. \[4 \cdot 63.5 = 254\]

    Find the necessary total. \[5\cdot70 = 350\]

    Find the difference of the totals. \[350-254 = 96\]

    The answer is 96.

    
    
16. 
    

    Question

    prevnext

    A student has taken 4 exams and gotten an average of 75.5. What score does the student need on the next exam to bring the average to 80? (All exams are equally weighted in the average.)

    
    

    Solution

    Show solution

    Find the current total. \[4 \cdot 75.5 = 302\]

    Find the necessary total. \[5\cdot80 = 400\]

    Find the difference of the totals. \[400-302 = 98\]

    The answer is 98.

    
    
17. 
    

    Question

    prevnext

    A student has taken 5 exams and gotten an average of 82.2. What score does the student need on the next exam to bring the average to 85? (All exams are equally weighted in the average.)

    
    

    Solution

    Show solution

    Find the current total. \[5 \cdot 82.2 = 411\]

    Find the necessary total. \[6\cdot85 = 510\]

    Find the difference of the totals. \[510-411 = 99\]

    The answer is 99.

    
    
18. 
    

    Question

    prevnext

    A student has taken 7 exams and gotten an average of 62. What score does the student need on the next exam to bring the average to 65? (All exams are equally weighted in the average.)

    
    

    Solution

    Show solution

    Find the current total. \[7 \cdot 62 = 434\]

    Find the necessary total. \[8\cdot65 = 520\]

    Find the difference of the totals. \[520-434 = 86\]

    The answer is 86.

    
    
19. 
    

    Question

    prevnext

    A student has taken 4 exams and gotten an average of 61.5. What score does the student need on the next exam to bring the average to 65? (All exams are equally weighted in the average.)

    
    

    Solution

    Show solution

    Find the current total. \[4 \cdot 61.5 = 246\]

    Find the necessary total. \[5\cdot65 = 325\]

    Find the difference of the totals. \[325-246 = 79\]

    The answer is 79.

    
    
20. 
    

    Question

    prevnext

    A student has taken 3 exams and gotten an average of 61. What score does the student need on the next exam to bring the average to 70? (All exams are equally weighted in the average.)

    
    

    Solution

    Show solution

    Find the current total. \[3 \cdot 61 = 183\]

    Find the necessary total. \[4\cdot70 = 280\]

    Find the difference of the totals. \[280-183 = 97\]

    The answer is 97.

    
    
21. 
    

    Question

    prevnext

    A lifeguard wants to get to a struggling swimmer as soon as possible. The lifeguard can run along the beach at 3.7 m/s and swim at 1.6 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(170-x)^2+20^2}\) meters.

    
    

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{3.7} + \frac{\sqrt{(170-x)^2+20^2}}{1.6}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.

    
    

    Solution

    Show solution

    Use a graphing utility to find the minimum.

    
    

    So the optimal value of \(x\) equals 160.41 in order to get to the struggling swimmer as quickly as possible.

    
    
22. 
    

    Question

    prevnext

    A lifeguard wants to get to a struggling swimmer as soon as possible. The lifeguard can run along the beach at 3 m/s and swim at 0.9 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(130-x)^2+30^2}\) meters.

    
    

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{3} + \frac{\sqrt{(130-x)^2+30^2}}{0.9}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.

    
    

    Solution

    Show solution

    Use a graphing utility to find the minimum.

    
    

    So the optimal value of \(x\) equals 120.57 in order to get to the struggling swimmer as quickly as possible.

    
    
23. 
    

    Question

    prevnext

    A lifeguard wants to get to a struggling swimmer as soon as possible. The lifeguard can run along the beach at 4 m/s and swim at 1.8 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(290-x)^2+10^2}\) meters.

    
    

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{4} + \frac{\sqrt{(290-x)^2+10^2}}{1.8}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.

    
    

    Solution

    Show solution

    Use a graphing utility to find the minimum.

    
    

    So the optimal value of \(x\) equals 284.96 in order to get to the struggling swimmer as quickly as possible.

    
    
24. 
    

    Question

    prevnext

    A lifeguard wants to get to a struggling swimmer as soon as possible. The lifeguard can run along the beach at 2.1 m/s and swim at 1.2 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(180-x)^2+60^2}\) meters.

    
    

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{2.1} + \frac{\sqrt{(180-x)^2+60^2}}{1.2}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.

    
    

    Solution

    Show solution

    Use a graphing utility to find the minimum.

    
    

    So the optimal value of \(x\) equals 138.22 in order to get to the struggling swimmer as quickly as possible.

    
    
25. 
    

    Question

    prevnext

    A lifeguard wants to get to a struggling swimmer as soon as possible. The lifeguard can run along the beach at 3.5 m/s and swim at 1.2 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(300-x)^2+60^2}\) meters.

    
    

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{3.5} + \frac{\sqrt{(300-x)^2+60^2}}{1.2}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.

    
    

    Solution

    Show solution

    Use a graphing utility to find the minimum.

    
    

    So the optimal value of \(x\) equals 278.1 in order to get to the struggling swimmer as quickly as possible.

    
    
26. 
    

    Question

    prevnext

    A lifeguard wants to get to a struggling swimmer as soon as possible. The lifeguard can run along the beach at 3.7 m/s and swim at 1 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(250-x)^2+50^2}\) meters.

    
    

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{3.7} + \frac{\sqrt{(250-x)^2+50^2}}{1}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.

    
    

    Solution

    Show solution

    Use a graphing utility to find the minimum.

    
    

    So the optimal value of \(x\) equals 235.96 in order to get to the struggling swimmer as quickly as possible.

    
    
27. 
    

    Question

    prevnext

    A lifeguard wants to get to a struggling swimmer as soon as possible. The lifeguard can run along the beach at 3.7 m/s and swim at 1.8 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(280-x)^2+40^2}\) meters.

    
    

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{3.7} + \frac{\sqrt{(280-x)^2+40^2}}{1.8}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.

    
    

    Solution

    Show solution

    Use a graphing utility to find the minimum.

    
    

    So the optimal value of \(x\) equals 257.73 in order to get to the struggling swimmer as quickly as possible.

    
    
28. 
    

    Question

    prevnext

    A lifeguard wants to get to a struggling swimmer as soon as possible. The lifeguard can run along the beach at 3.9 m/s and swim at 1.3 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(240-x)^2+30^2}\) meters.

    
    

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{3.9} + \frac{\sqrt{(240-x)^2+30^2}}{1.3}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.

    
    

    Solution

    Show solution

    Use a graphing utility to find the minimum.

    
    

    So the optimal value of \(x\) equals 229.39 in order to get to the struggling swimmer as quickly as possible.

    
    
29. 
    

    Question

    prevnext

    A lifeguard wants to get to a struggling swimmer as soon as possible. The lifeguard can run along the beach at 3.9 m/s and swim at 1.1 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(170-x)^2+30^2}\) meters.

    
    

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{3.9} + \frac{\sqrt{(170-x)^2+30^2}}{1.1}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.

    
    

    Solution

    Show solution

    Use a graphing utility to find the minimum.

    
    

    So the optimal value of \(x\) equals 161.18 in order to get to the struggling swimmer as quickly as possible.

    
    
30. 
    

    Question

    prevnext

    A lifeguard wants to get to a struggling swimmer as soon as possible. The lifeguard can run along the beach at 3.3 m/s and swim at 1.6 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(240-x)^2+30^2}\) meters.

    
    

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{3.3} + \frac{\sqrt{(240-x)^2+30^2}}{1.6}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.

    
    

    Solution

    Show solution

    Use a graphing utility to find the minimum.

    
    

    So the optimal value of \(x\) equals 223.37 in order to get to the struggling swimmer as quickly as possible.

    
    
31. 
    

    Question

    prevnext

    The figure below summarizes the empirical rule. If a normally-distributed population has a mean length of \(\mu=800\) cm and a standard deviation of \(\sigma=15\) cm, then what is the probability an individual has a length between 785 cm and 830 cm?

    The tolerance is \(\pm 0.01\) cm.

    
    

    
    

    Solution

    Show solution

    Fill in the values in the diagram.

    
    

    You can find the \(z\) scores if that is helpful.

    \[z_1 ~=~ \frac{785-800}{15} ~=~ -1\] \[z_2 ~=~ \frac{830-800}{15} ~=~ 2\]

    Add up the areas.

    \[P(785\le X \le 830) ~=~ P(-1\le Z \le 2) ~\approx~ 0.82\]

    
    
32. 
    

    Question

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    The figure below summarizes the empirical rule. If a normally-distributed population has a mean length of \(\mu=750\) cm and a standard deviation of \(\sigma=20\) cm, then what is the probability an individual has a length between 690 cm and 750 cm?

    The tolerance is \(\pm 0.01\) cm.

    
    

    
    

    Solution

    Show solution

    Fill in the values in the diagram.

    
    

    You can find the \(z\) scores if that is helpful.

    \[z_1 ~=~ \frac{690-750}{20} ~=~ -3\] \[z_2 ~=~ \frac{750-750}{20} ~=~ 0\]

    Add up the areas.

    \[P(690\le X \le 750) ~=~ P(-3\le Z \le 0) ~\approx~ 0.5\]

    
    
33. 
    

    Question

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    The figure below summarizes the empirical rule. If a normally-distributed population has a mean length of \(\mu=250\) cm and a standard deviation of \(\sigma=5\) cm, then what is the probability an individual has a length between 240 cm and 260 cm?

    The tolerance is \(\pm 0.01\) cm.

    
    

    
    

    Solution

    Show solution

    Fill in the values in the diagram.

    
    

    You can find the \(z\) scores if that is helpful.

    \[z_1 ~=~ \frac{240-250}{5} ~=~ -2\] \[z_2 ~=~ \frac{260-250}{5} ~=~ 2\]

    Add up the areas.

    \[P(240\le X \le 260) ~=~ P(-2\le Z \le 2) ~\approx~ 0.95\]

    
    
34. 
    

    Question

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    The figure below summarizes the empirical rule. If a normally-distributed population has a mean length of \(\mu=300\) cm and a standard deviation of \(\sigma=30\) cm, then what is the probability an individual has a length between 300 cm and 360 cm?

    The tolerance is \(\pm 0.01\) cm.

    
    

    
    

    Solution

    Show solution

    Fill in the values in the diagram.

    
    

    You can find the \(z\) scores if that is helpful.

    \[z_1 ~=~ \frac{300-300}{30} ~=~ 0\] \[z_2 ~=~ \frac{360-300}{30} ~=~ 2\]

    Add up the areas.

    \[P(300\le X \le 360) ~=~ P(0\le Z \le 2) ~\approx~ 0.48\]

    
    
35. 
    

    Question

    prevnext

    The figure below summarizes the empirical rule. If a normally-distributed population has a mean length of \(\mu=700\) cm and a standard deviation of \(\sigma=2\) cm, then what is the probability an individual has a length between 698 cm and 702 cm?

    The tolerance is \(\pm 0.01\) cm.

    
    

    
    

    Solution

    Show solution

    Fill in the values in the diagram.

    
    

    You can find the \(z\) scores if that is helpful.

    \[z_1 ~=~ \frac{698-700}{2} ~=~ -1\] \[z_2 ~=~ \frac{702-700}{2} ~=~ 1\]

    Add up the areas.

    \[P(698\le X \le 702) ~=~ P(-1\le Z \le 1) ~\approx~ 0.68\]

    
    
36. 
    

    Question

    prevnext

    The figure below summarizes the empirical rule. If a normally-distributed population has a mean length of \(\mu=300\) cm and a standard deviation of \(\sigma=25\) cm, then what is the probability an individual has a length between 250 cm and 375 cm?

    The tolerance is \(\pm 0.01\) cm.

    
    

    
    

    Solution

    Show solution

    Fill in the values in the diagram.

    
    

    You can find the \(z\) scores if that is helpful.

    \[z_1 ~=~ \frac{250-300}{25} ~=~ -2\] \[z_2 ~=~ \frac{375-300}{25} ~=~ 3\]

    Add up the areas.

    \[P(250\le X \le 375) ~=~ P(-2\le Z \le 3) ~\approx~ 0.98\]

    
    
37. 
    

    Question

    prevnext

    The figure below summarizes the empirical rule. If a normally-distributed population has a mean length of \(\mu=600\) cm and a standard deviation of \(\sigma=2\) cm, then what is the probability an individual has a length between 594 cm and 606 cm?

    The tolerance is \(\pm 0.01\) cm.

    
    

    
    

    Solution

    Show solution

    Fill in the values in the diagram.

    
    

    You can find the \(z\) scores if that is helpful.

    \[z_1 ~=~ \frac{594-600}{2} ~=~ -3\] \[z_2 ~=~ \frac{606-600}{2} ~=~ 3\]

    Add up the areas.

    \[P(594\le X \le 606) ~=~ P(-3\le Z \le 3) ~\approx~ 1\]

    
    
38. 
    

    Question

    prevnext

    The figure below summarizes the empirical rule. If a normally-distributed population has a mean length of \(\mu=200\) cm and a standard deviation of \(\sigma=5\) cm, then what is the probability an individual has a length between 190 cm and 195 cm?

    The tolerance is \(\pm 0.01\) cm.

    
    

    
    

    Solution

    Show solution

    Fill in the values in the diagram.

    
    

    You can find the \(z\) scores if that is helpful.

    \[z_1 ~=~ \frac{190-200}{5} ~=~ -2\] \[z_2 ~=~ \frac{195-200}{5} ~=~ -1\]

    Add up the areas.

    \[P(190\le X \le 195) ~=~ P(-2\le Z \le -1) ~\approx~ 0.14\]

    
    
39. 
    

    Question

    prevnext

    The figure below summarizes the empirical rule. If a normally-distributed population has a mean length of \(\mu=450\) cm and a standard deviation of \(\sigma=20\) cm, then what is the probability an individual has a length between 490 cm and 510 cm?

    The tolerance is \(\pm 0.01\) cm.

    
    

    
    

    Solution

    Show solution

    Fill in the values in the diagram.

    
    

    You can find the \(z\) scores if that is helpful.

    \[z_1 ~=~ \frac{490-450}{20} ~=~ 2\] \[z_2 ~=~ \frac{510-450}{20} ~=~ 3\]

    Add up the areas.

    \[P(490\le X \le 510) ~=~ P(2\le Z \le 3) ~\approx~ 0.02\]

    
    
40. 
    

    Question

    prevnext

    The figure below summarizes the empirical rule. If a normally-distributed population has a mean length of \(\mu=650\) cm and a standard deviation of \(\sigma=15\) cm, then what is the probability an individual has a length between 620 cm and 665 cm?

    The tolerance is \(\pm 0.01\) cm.

    
    

    
    

    Solution

    Show solution

    Fill in the values in the diagram.

    
    

    You can find the \(z\) scores if that is helpful.

    \[z_1 ~=~ \frac{620-650}{15} ~=~ -2\] \[z_2 ~=~ \frac{665-650}{15} ~=~ 1\]

    Add up the areas.

    \[P(620\le X \le 665) ~=~ P(-2\le Z \le 1) ~\approx~ 0.82\]

    
    
41. 
    

    Question

    prevnext

    Below is a 2-way table expressing joint probabilities.

    	A	not A
    B	33%	42%
    not B	11%	14%

    So,

    \[\begin{align} P(A \text{ and }B) &= 0.33 \\ P(A \text{ and not }B) &= 0.11 \\ P([\text{not }A] \text{ and }B) &= 0.42 \\ P([\text{not }A] \text{ and [not }B]) &= 0.14 \\ \end{align}\]

    Are events \(A\) and \(B\) independent?

    
    
    1. Yes, events A and B are independent.
    2. No, events A and B are dependent.
    
    

    Solution

    Show solution

    First, find \(P(A)\). \[P(A) ~=~ 0.33+0.11 ~=~0.44\]

    Find \(P(B)\). \[P(B) ~=~ 0.33+0.42 ~=~0.75\]

    Next, find \(P(A\text{ given }B)\). \[P(A|B) ~=~ \frac{P(A\text{ and }B)}{P(B)} ~=~ \frac{0.33}{0.75} ~=~ 0.44\]

    Compare \(P(A)\) and \(P(A|B)\). The events \(A\) and \(B\) are independent if and only if \(P(A)=P(A|B)\). \[0.44 ~=~ 0.44\] So, \(A\) and \(B\) are independent.

    
    
42. 
    

    Question

    prevnext

    Below is a 2-way table expressing joint probabilities.

    	A	not A
    B	27%	48%
    not B	9%	16%

    So,

    \[\begin{align} P(A \text{ and }B) &= 0.27 \\ P(A \text{ and not }B) &= 0.09 \\ P([\text{not }A] \text{ and }B) &= 0.48 \\ P([\text{not }A] \text{ and [not }B]) &= 0.16 \\ \end{align}\]

    Are events \(A\) and \(B\) independent?

    
    
    1. Yes, events A and B are independent.
    2. No, events A and B are dependent.
    
    

    Solution

    Show solution

    First, find \(P(A)\). \[P(A) ~=~ 0.27+0.09 ~=~0.36\]

    Find \(P(B)\). \[P(B) ~=~ 0.27+0.48 ~=~0.75\]

    Next, find \(P(A\text{ given }B)\). \[P(A|B) ~=~ \frac{P(A\text{ and }B)}{P(B)} ~=~ \frac{0.27}{0.75} ~=~ 0.36\]

    Compare \(P(A)\) and \(P(A|B)\). The events \(A\) and \(B\) are independent if and only if \(P(A)=P(A|B)\). \[0.36 ~=~ 0.36\] So, \(A\) and \(B\) are independent.

    
    
43. 
    

    Question

    prevnext

    Below is a 2-way table expressing joint probabilities.

    	A	not A
    B	7%	13%
    not B	28%	52%

    So,

    \[\begin{align} P(A \text{ and }B) &= 0.07 \\ P(A \text{ and not }B) &= 0.28 \\ P([\text{not }A] \text{ and }B) &= 0.13 \\ P([\text{not }A] \text{ and [not }B]) &= 0.52 \\ \end{align}\]

    Are events \(A\) and \(B\) independent?

    
    
    1. Yes, events A and B are independent.
    2. No, events A and B are dependent.
    
    

    Solution

    Show solution

    First, find \(P(A)\). \[P(A) ~=~ 0.07+0.28 ~=~0.35\]

    Find \(P(B)\). \[P(B) ~=~ 0.07+0.13 ~=~0.2\]

    Next, find \(P(A\text{ given }B)\). \[P(A|B) ~=~ \frac{P(A\text{ and }B)}{P(B)} ~=~ \frac{0.07}{0.2} ~=~ 0.35\]

    Compare \(P(A)\) and \(P(A|B)\). The events \(A\) and \(B\) are independent if and only if \(P(A)=P(A|B)\). \[0.35 ~=~ 0.35\] So, \(A\) and \(B\) are independent.

    
    
44. 
    

    Question

    prevnext

    Below is a 2-way table expressing joint probabilities.

    	A	not A
    B	33%	28%
    not B	17%	22%

    So,

    \[\begin{align} P(A \text{ and }B) &= 0.33 \\ P(A \text{ and not }B) &= 0.17 \\ P([\text{not }A] \text{ and }B) &= 0.28 \\ P([\text{not }A] \text{ and [not }B]) &= 0.22 \\ \end{align}\]

    Are events \(A\) and \(B\) independent?

    
    
    1. Yes, events A and B are independent.
    2. No, events A and B are dependent.
    
    

    Solution

    Show solution

    First, find \(P(A)\). \[P(A) ~=~ 0.33+0.17 ~=~0.5\]

    Find \(P(B)\). \[P(B) ~=~ 0.33+0.28 ~=~0.61\]

    Next, find \(P(A\text{ given }B)\). \[P(A|B) ~=~ \frac{P(A\text{ and }B)}{P(B)} ~=~ \frac{0.33}{0.61} ~=~ 0.5409836\]

    Compare \(P(A)\) and \(P(A|B)\). The events \(A\) and \(B\) are independent if and only if \(P(A)=P(A|B)\). \[0.5 ~\ne~ 0.5409836\] So, \(A\) and \(B\) are dependent.

    
    
45. 
    

    Question

    prevnext

    Below is a 2-way table expressing joint probabilities.

    	A	not A
    B	19%	16%
    not B	31%	34%

    So,

    \[\begin{align} P(A \text{ and }B) &= 0.19 \\ P(A \text{ and not }B) &= 0.31 \\ P([\text{not }A] \text{ and }B) &= 0.16 \\ P([\text{not }A] \text{ and [not }B]) &= 0.34 \\ \end{align}\]

    Are events \(A\) and \(B\) independent?

    
    
    1. Yes, events A and B are independent.
    2. No, events A and B are dependent.
    
    

    Solution

    Show solution

    First, find \(P(A)\). \[P(A) ~=~ 0.19+0.31 ~=~0.5\]

    Find \(P(B)\). \[P(B) ~=~ 0.19+0.16 ~=~0.35\]

    Next, find \(P(A\text{ given }B)\). \[P(A|B) ~=~ \frac{P(A\text{ and }B)}{P(B)} ~=~ \frac{0.19}{0.35} ~=~ 0.5428571\]

    Compare \(P(A)\) and \(P(A|B)\). The events \(A\) and \(B\) are independent if and only if \(P(A)=P(A|B)\). \[0.5 ~\ne~ 0.5428571\] So, \(A\) and \(B\) are dependent.

    
    
46. 
    

    Question

    prevnext

    Below is a 2-way table expressing joint probabilities.

    	A	not A
    B	15%	5%
    not B	60%	20%

    So,

    \[\begin{align} P(A \text{ and }B) &= 0.15 \\ P(A \text{ and not }B) &= 0.6 \\ P([\text{not }A] \text{ and }B) &= 0.05 \\ P([\text{not }A] \text{ and [not }B]) &= 0.2 \\ \end{align}\]

    Are events \(A\) and \(B\) independent?

    
    
    1. Yes, events A and B are independent.
    2. No, events A and B are dependent.
    
    

    Solution

    Show solution

    First, find \(P(A)\). \[P(A) ~=~ 0.15+0.6 ~=~0.75\]

    Find \(P(B)\). \[P(B) ~=~ 0.15+0.05 ~=~0.2\]

    Next, find \(P(A\text{ given }B)\). \[P(A|B) ~=~ \frac{P(A\text{ and }B)}{P(B)} ~=~ \frac{0.15}{0.2} ~=~ 0.75\]

    Compare \(P(A)\) and \(P(A|B)\). The events \(A\) and \(B\) are independent if and only if \(P(A)=P(A|B)\). \[0.75 ~=~ 0.75\] So, \(A\) and \(B\) are independent.

    
    
47. 
    

    Question

    prevnext

    Below is a 2-way table expressing joint probabilities.

    	A	not A
    B	30%	49%
    not B	10%	11%

    So,

    \[\begin{align} P(A \text{ and }B) &= 0.3 \\ P(A \text{ and not }B) &= 0.1 \\ P([\text{not }A] \text{ and }B) &= 0.49 \\ P([\text{not }A] \text{ and [not }B]) &= 0.11 \\ \end{align}\]

    Are events \(A\) and \(B\) independent?

    
    
    1. Yes, events A and B are independent.
    2. No, events A and B are dependent.
    
    

    Solution

    Show solution

    First, find \(P(A)\). \[P(A) ~=~ 0.3+0.1 ~=~0.4\]

    Find \(P(B)\). \[P(B) ~=~ 0.3+0.49 ~=~0.79\]

    Next, find \(P(A\text{ given }B)\). \[P(A|B) ~=~ \frac{P(A\text{ and }B)}{P(B)} ~=~ \frac{0.3}{0.79} ~=~ 0.3797468\]

    Compare \(P(A)\) and \(P(A|B)\). The events \(A\) and \(B\) are independent if and only if \(P(A)=P(A|B)\). \[0.4 ~\ne~ 0.3797468\] So, \(A\) and \(B\) are dependent.

    
    
48. 
    

    Question

    prevnext

    Below is a 2-way table expressing joint probabilities.

    	A	not A
    B	6%	14%
    not B	24%	56%

    So,

    \[\begin{align} P(A \text{ and }B) &= 0.06 \\ P(A \text{ and not }B) &= 0.24 \\ P([\text{not }A] \text{ and }B) &= 0.14 \\ P([\text{not }A] \text{ and [not }B]) &= 0.56 \\ \end{align}\]

    Are events \(A\) and \(B\) independent?

    
    
    1. Yes, events A and B are independent.
    2. No, events A and B are dependent.
    
    

    Solution

    Show solution

    First, find \(P(A)\). \[P(A) ~=~ 0.06+0.24 ~=~0.3\]

    Find \(P(B)\). \[P(B) ~=~ 0.06+0.14 ~=~0.2\]

    Next, find \(P(A\text{ given }B)\). \[P(A|B) ~=~ \frac{P(A\text{ and }B)}{P(B)} ~=~ \frac{0.06}{0.2} ~=~ 0.3\]

    Compare \(P(A)\) and \(P(A|B)\). The events \(A\) and \(B\) are independent if and only if \(P(A)=P(A|B)\). \[0.3 ~=~ 0.3\] So, \(A\) and \(B\) are independent.

    
    
49. 
    

    Question

    prevnext

    Below is a 2-way table expressing joint probabilities.

    	A	not A
    B	36%	39%
    not B	12%	13%

    So,

    \[\begin{align} P(A \text{ and }B) &= 0.36 \\ P(A \text{ and not }B) &= 0.12 \\ P([\text{not }A] \text{ and }B) &= 0.39 \\ P([\text{not }A] \text{ and [not }B]) &= 0.13 \\ \end{align}\]

    Are events \(A\) and \(B\) independent?

    
    
    1. Yes, events A and B are independent.
    2. No, events A and B are dependent.
    
    

    Solution

    Show solution

    First, find \(P(A)\). \[P(A) ~=~ 0.36+0.12 ~=~0.48\]

    Find \(P(B)\). \[P(B) ~=~ 0.36+0.39 ~=~0.75\]

    Next, find \(P(A\text{ given }B)\). \[P(A|B) ~=~ \frac{P(A\text{ and }B)}{P(B)} ~=~ \frac{0.36}{0.75} ~=~ 0.48\]

    Compare \(P(A)\) and \(P(A|B)\). The events \(A\) and \(B\) are independent if and only if \(P(A)=P(A|B)\). \[0.48 ~=~ 0.48\] So, \(A\) and \(B\) are independent.

    
    
50. 
    

    Question

    prevnext

    Below is a 2-way table expressing joint probabilities.

    	A	not A
    B	12%	11%
    not B	36%	41%

    So,

    \[\begin{align} P(A \text{ and }B) &= 0.12 \\ P(A \text{ and not }B) &= 0.36 \\ P([\text{not }A] \text{ and }B) &= 0.11 \\ P([\text{not }A] \text{ and [not }B]) &= 0.41 \\ \end{align}\]

    Are events \(A\) and \(B\) independent?

    
    
    1. Yes, events A and B are independent.
    2. No, events A and B are dependent.
    
    

    Solution

    Show solution

    First, find \(P(A)\). \[P(A) ~=~ 0.12+0.36 ~=~0.48\]

    Find \(P(B)\). \[P(B) ~=~ 0.12+0.11 ~=~0.23\]

    Next, find \(P(A\text{ given }B)\). \[P(A|B) ~=~ \frac{P(A\text{ and }B)}{P(B)} ~=~ \frac{0.12}{0.23} ~=~ 0.5217391\]

    Compare \(P(A)\) and \(P(A|B)\). The events \(A\) and \(B\) are independent if and only if \(P(A)=P(A|B)\). \[0.48 ~\ne~ 0.5217391\] So, \(A\) and \(B\) are dependent.

    
    
51. 
    

    Question

    prevnext

    Below are 4 exponential functions.

    \[\begin{align} A(x) ~&=~ 10.7\cdot\left(\frac{37}{17}\right)^{-1.57x} \\\\ B(x) ~&=~ 92.4\cdot\left(\frac{23}{5}\right)^{0.89x} \\\\ C(x) ~&=~ 55.6\cdot\left(\frac{3}{31}\right)^{-1.38x} \\\\ D(x) ~&=~ 55.4\cdot\left(\frac{13}{29}\right)^{-0.38x} \end{align}\]

    Which function represents exponential decay?

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    There two ways \(f(x)=f_0\cdot M^{kx}\), with \(f_0>0\), can represent exponential decay:

    • \(M>1\) and \(k<0\)
    • \(M<1\) and \(k>0\)

    In this case, function \(A\) has \(M=37/17\) and \(k=-1.57\). So, in this case \(M > 1\) and \(k < 0\), giving exponential decay.

    
    
52. 
    

    Question

    prevnext

    Below are 4 exponential functions.

    \[\begin{align} A(x) ~&=~ 22.2\cdot\left(\frac{11}{5}\right)^{0.44x} \\\\ B(x) ~&=~ 73.3\cdot\left(\frac{3}{23}\right)^{0.58x} \\\\ C(x) ~&=~ 17.7\cdot\left(\frac{17}{2}\right)^{0.81x} \\\\ D(x) ~&=~ 53.3\cdot\left(\frac{13}{19}\right)^{-0.42x} \end{align}\]

    Which function represents exponential decay?

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    There two ways \(f(x)=f_0\cdot M^{kx}\), with \(f_0>0\), can represent exponential decay:

    • \(M>1\) and \(k<0\)
    • \(M<1\) and \(k>0\)

    In this case, function \(B\) has \(M=3/23\) and \(k=0.58\). So, in this case \(M < 1\) and \(k > 0\), giving exponential decay.

    
    
53. 
    

    Question

    prevnext

    Below are 4 exponential functions.

    \[\begin{align} A(x) ~&=~ 33.3\cdot\left(\frac{13}{5}\right)^{-1.21x} \\\\ B(x) ~&=~ 82.9\cdot\left(\frac{3}{31}\right)^{-1.2x} \\\\ C(x) ~&=~ 27\cdot\left(\frac{37}{17}\right)^{0.26x} \\\\ D(x) ~&=~ 56.4\cdot\left(\frac{7}{23}\right)^{-0.71x} \end{align}\]

    Which function represents exponential decay?

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    There two ways \(f(x)=f_0\cdot M^{kx}\), with \(f_0>0\), can represent exponential decay:

    • \(M>1\) and \(k<0\)
    • \(M<1\) and \(k>0\)

    In this case, function \(A\) has \(M=13/5\) and \(k=-1.21\). So, in this case \(M > 1\) and \(k < 0\), giving exponential decay.

    
    
54. 
    

    Question

    prevnext

    Below are 4 exponential functions.

    \[\begin{align} A(x) ~&=~ 42.1\cdot\left(\frac{2}{23}\right)^{-1.9x} \\\\ B(x) ~&=~ 95.5\cdot\left(\frac{7}{3}\right)^{1.9x} \\\\ C(x) ~&=~ 45.1\cdot\left(\frac{29}{37}\right)^{0.67x} \\\\ D(x) ~&=~ 55.1\cdot\left(\frac{13}{17}\right)^{-1.78x} \end{align}\]

    Which function represents exponential decay?

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    There two ways \(f(x)=f_0\cdot M^{kx}\), with \(f_0>0\), can represent exponential decay:

    • \(M>1\) and \(k<0\)
    • \(M<1\) and \(k>0\)

    In this case, function \(C\) has \(M=29/37\) and \(k=0.67\). So, in this case \(M < 1\) and \(k > 0\), giving exponential decay.

    
    
55. 
    

    Question

    prevnext

    Below are 4 exponential functions.

    \[\begin{align} A(x) ~&=~ 77\cdot\left(\frac{31}{3}\right)^{1.82x} \\\\ B(x) ~&=~ 47.5\cdot\left(\frac{17}{5}\right)^{1.98x} \\\\ C(x) ~&=~ 84.6\cdot\left(\frac{29}{19}\right)^{1.69x} \\\\ D(x) ~&=~ 21\cdot\left(\frac{23}{7}\right)^{-1.13x} \end{align}\]

    Which function represents exponential decay?

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    There two ways \(f(x)=f_0\cdot M^{kx}\), with \(f_0>0\), can represent exponential decay:

    • \(M>1\) and \(k<0\)
    • \(M<1\) and \(k>0\)

    In this case, function \(D\) has \(M=23/7\) and \(k=-1.13\). So, in this case \(M > 1\) and \(k < 0\), giving exponential decay.

    
    
56. 
    

    Question

    prevnext

    Below are 4 exponential functions.

    \[\begin{align} A(x) ~&=~ 94.1\cdot\left(\frac{7}{2}\right)^{0.22x} \\\\ B(x) ~&=~ 10.8\cdot\left(\frac{17}{23}\right)^{0.34x} \\\\ C(x) ~&=~ 81.8\cdot\left(\frac{29}{37}\right)^{-0.72x} \\\\ D(x) ~&=~ 71.7\cdot\left(\frac{31}{19}\right)^{1.25x} \end{align}\]

    Which function represents exponential decay?

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    There two ways \(f(x)=f_0\cdot M^{kx}\), with \(f_0>0\), can represent exponential decay:

    • \(M>1\) and \(k<0\)
    • \(M<1\) and \(k>0\)

    In this case, function \(B\) has \(M=17/23\) and \(k=0.34\). So, in this case \(M < 1\) and \(k > 0\), giving exponential decay.

    
    
57. 
    

    Question

    prevnext

    Below are 4 exponential functions.

    \[\begin{align} A(x) ~&=~ 78.5\cdot\left(\frac{19}{23}\right)^{1.37x} \\\\ B(x) ~&=~ 89.7\cdot\left(\frac{29}{37}\right)^{-0.83x} \\\\ C(x) ~&=~ 32.3\cdot\left(\frac{5}{2}\right)^{0.72x} \\\\ D(x) ~&=~ 37.9\cdot\left(\frac{7}{31}\right)^{-0.29x} \end{align}\]

    Which function represents exponential decay?

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    There two ways \(f(x)=f_0\cdot M^{kx}\), with \(f_0>0\), can represent exponential decay:

    • \(M>1\) and \(k<0\)
    • \(M<1\) and \(k>0\)

    In this case, function \(A\) has \(M=19/23\) and \(k=1.37\). So, in this case \(M < 1\) and \(k > 0\), giving exponential decay.

    
    
58. 
    

    Question

    prevnext

    Below are 4 exponential functions.

    \[\begin{align} A(x) ~&=~ 88.5\cdot\left(\frac{7}{29}\right)^{0.24x} \\\\ B(x) ~&=~ 48.3\cdot\left(\frac{5}{31}\right)^{-0.27x} \\\\ C(x) ~&=~ 72.3\cdot\left(\frac{13}{23}\right)^{-1.77x} \\\\ D(x) ~&=~ 76.4\cdot\left(\frac{11}{19}\right)^{-1.69x} \end{align}\]

    Which function represents exponential decay?

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    There two ways \(f(x)=f_0\cdot M^{kx}\), with \(f_0>0\), can represent exponential decay:

    • \(M>1\) and \(k<0\)
    • \(M<1\) and \(k>0\)

    In this case, function \(A\) has \(M=7/29\) and \(k=0.24\). So, in this case \(M < 1\) and \(k > 0\), giving exponential decay.

    
    
59. 
    

    Question

    prevnext

    Below are 4 exponential functions.

    \[\begin{align} A(x) ~&=~ 73.9\cdot\left(\frac{5}{37}\right)^{-1.22x} \\\\ B(x) ~&=~ 32.3\cdot\left(\frac{29}{19}\right)^{0.34x} \\\\ C(x) ~&=~ 90.6\cdot\left(\frac{13}{17}\right)^{1.2x} \\\\ D(x) ~&=~ 40.7\cdot\left(\frac{31}{3}\right)^{1.5x} \end{align}\]

    Which function represents exponential decay?

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    There two ways \(f(x)=f_0\cdot M^{kx}\), with \(f_0>0\), can represent exponential decay:

    • \(M>1\) and \(k<0\)
    • \(M<1\) and \(k>0\)

    In this case, function \(C\) has \(M=13/17\) and \(k=1.2\). So, in this case \(M < 1\) and \(k > 0\), giving exponential decay.

    
    
60. 
    

    Question

    prevnext

    Below are 4 exponential functions.

    \[\begin{align} A(x) ~&=~ 93.4\cdot\left(\frac{19}{13}\right)^{1.4x} \\\\ B(x) ~&=~ 68.1\cdot\left(\frac{7}{37}\right)^{-1.94x} \\\\ C(x) ~&=~ 33.1\cdot\left(\frac{23}{3}\right)^{1.33x} \\\\ D(x) ~&=~ 69.9\cdot\left(\frac{11}{31}\right)^{0.19x} \end{align}\]

    Which function represents exponential decay?

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    There two ways \(f(x)=f_0\cdot M^{kx}\), with \(f_0>0\), can represent exponential decay:

    • \(M>1\) and \(k<0\)
    • \(M<1\) and \(k>0\)

    In this case, function \(D\) has \(M=11/31\) and \(k=0.19\). So, in this case \(M < 1\) and \(k > 0\), giving exponential decay.

    
    
61. 
    

    Question

    prevnext

    Let function \(f\) be defined with the algebraic formula below: \[f(x) = \frac{x}{8}-3\]

    Find an expression for the inverse function \(f^{-1}\).

    
    
    1. \(f^{-1}(x)~=~\frac{x}{3}+8\)
    2. \(f^{-1}(x)~=~8(x-3)\)
    3. \(f^{-1}(x)~=~8(x+3)\)
    4. \(f^{-1}(x)~=~3x+8\)
    
    

    Solution

    Show solution

    The original function tells us to divide by 8 and then subtract 3.

    The inverse function does the inverse operations in the reverse order, so it must add 3 and then multiply by 8.

    \[f^{-1}(x)=8(x+3)\]

    
    
62. 
    

    Question

    prevnext

    Let function \(f\) be defined with the algebraic formula below: \[f(x) = 8x+5\]

    Find an expression for the inverse function \(f^{-1}\).

    
    
    1. \(f^{-1}(x)~=~8(x+5)\)
    2. \(f^{-1}(x)~=~\frac{x}{5}+8\)
    3. \(f^{-1}(x)~=~\frac{x-5}{8}\)
    4. \(f^{-1}(x)~=~\frac{x+5}{8}\)
    
    

    Solution

    Show solution

    The original function tells us to multiply by 8 and then add 5.

    The inverse function does the inverse operations in the reverse order, so it must subtract 5 and then divide by 8.

    \[f^{-1}(x)=\frac{x-5}{8}\]

    
    
63. 
    

    Question

    prevnext

    Let function \(f\) be defined with the algebraic formula below: \[f(x) = 3x+4\]

    Find an expression for the inverse function \(f^{-1}\).

    
    
    1. \(f^{-1}(x)~=~\frac{x}{4}-3\)
    2. \(f^{-1}(x)~=~\frac{x-4}{3}\)
    3. \(f^{-1}(x)~=~4x-3\)
    4. \(f^{-1}(x)~=~4x+3\)
    
    

    Solution

    Show solution

    The original function tells us to multiply by 3 and then add 4.

    The inverse function does the inverse operations in the reverse order, so it must subtract 4 and then divide by 3.

    \[f^{-1}(x)=\frac{x-4}{3}\]

    
    
64. 
    

    Question

    prevnext

    Let function \(f\) be defined with the algebraic formula below: \[f(x) = \frac{x}{5}-4\]

    Find an expression for the inverse function \(f^{-1}\).

    
    
    1. \(f^{-1}(x)~=~5(x+4)\)
    2. \(f^{-1}(x)~=~\frac{x}{4}-5\)
    3. \(f^{-1}(x)~=~\frac{x-4}{5}\)
    4. \(f^{-1}(x)~=~4x-5\)
    
    

    Solution

    Show solution

    The original function tells us to divide by 5 and then subtract 4.

    The inverse function does the inverse operations in the reverse order, so it must add 4 and then multiply by 5.

    \[f^{-1}(x)=5(x+4)\]

    
    
65. 
    

    Question

    prevnext

    Let function \(f\) be defined with the algebraic formula below: \[f(x) = 4x-3\]

    Find an expression for the inverse function \(f^{-1}\).

    
    
    1. \(f^{-1}(x)~=~4(x-3)\)
    2. \(f^{-1}(x)~=~\frac{x+3}{4}\)
    3. \(f^{-1}(x)~=~\frac{x}{3}-4\)
    4. \(f^{-1}(x)~=~\frac{x}{3}+4\)
    
    

    Solution

    Show solution

    The original function tells us to multiply by 4 and then subtract 3.

    The inverse function does the inverse operations in the reverse order, so it must add 3 and then divide by 4.

    \[f^{-1}(x)=\frac{x+3}{4}\]

    
    
66. 
    

    Question

    prevnext

    Let function \(f\) be defined with the algebraic formula below: \[f(x) = 6x-7\]

    Find an expression for the inverse function \(f^{-1}\).

    
    
    1. \(f^{-1}(x)~=~\frac{x+7}{6}\)
    2. \(f^{-1}(x)~=~\frac{x}{7}-6\)
    3. \(f^{-1}(x)~=~\frac{x-7}{6}\)
    4. \(f^{-1}(x)~=~6(x-7)\)
    
    

    Solution

    Show solution

    The original function tells us to multiply by 6 and then subtract 7.

    The inverse function does the inverse operations in the reverse order, so it must add 7 and then divide by 6.

    \[f^{-1}(x)=\frac{x+7}{6}\]

    
    
67. 
    

    Question

    prevnext

    Let function \(f\) be defined with the algebraic formula below: \[f(x) = \frac{x-9}{5}\]

    Find an expression for the inverse function \(f^{-1}\).

    
    
    1. \(f^{-1}(x)~=~9(x-5)\)
    2. \(f^{-1}(x)~=~\frac{x+5}{9}\)
    3. \(f^{-1}(x)~=~\frac{x-5}{9}\)
    4. \(f^{-1}(x)~=~5x+9\)
    
    

    Solution

    Show solution

    The original function tells us to subtract 9 and then divide by 5.

    The inverse function does the inverse operations in the reverse order, so it must multiply by 5 and then add 9.

    \[f^{-1}(x)=5x+9\]

    
    
68. 
    

    Question

    prevnext

    Let function \(f\) be defined with the algebraic formula below: \[f(x) = \frac{x}{3}-9\]

    Find an expression for the inverse function \(f^{-1}\).

    
    
    1. \(f^{-1}(x)~=~3(x+9)\)
    2. \(f^{-1}(x)~=~\frac{x}{9}-3\)
    3. \(f^{-1}(x)~=~3(x-9)\)
    4. \(f^{-1}(x)~=~9x-3\)
    
    

    Solution

    Show solution

    The original function tells us to divide by 3 and then subtract 9.

    The inverse function does the inverse operations in the reverse order, so it must add 9 and then multiply by 3.

    \[f^{-1}(x)=3(x+9)\]

    
    
69. 
    

    Question

    prevnext

    Let function \(f\) be defined with the algebraic formula below: \[f(x) = \frac{x}{9}+5\]

    Find an expression for the inverse function \(f^{-1}\).

    
    
    1. \(f^{-1}(x)~=~\frac{x-5}{9}\)
    2. \(f^{-1}(x)~=~9(x-5)\)
    3. \(f^{-1}(x)~=~\frac{x}{5}-9\)
    4. \(f^{-1}(x)~=~9(x+5)\)
    
    

    Solution

    Show solution

    The original function tells us to divide by 9 and then add 5.

    The inverse function does the inverse operations in the reverse order, so it must subtract 5 and then multiply by 9.

    \[f^{-1}(x)=9(x-5)\]

    
    
70. 
    

    Question

    prevnext

    Let function \(f\) be defined with the algebraic formula below: \[f(x) = 6(x+5)\]

    Find an expression for the inverse function \(f^{-1}\).

    
    
    1. \(f^{-1}(x)~=~\frac{x}{6}+5\)
    2. \(f^{-1}(x)~=~\frac{x}{6}-5\)
    3. \(f^{-1}(x)~=~\frac{x+6}{5}\)
    4. \(f^{-1}(x)~=~\frac{x-6}{5}\)
    
    

    Solution

    Show solution

    The original function tells us to add 5 and then multiply by 6.

    The inverse function does the inverse operations in the reverse order, so it must divide by 6 and then subtract 5.

    \[f^{-1}(x)=\frac{x}{6}-5\]

    
    
71. 
    

    Question

    prevnext

    Which of the following has \(9+8i\) as a solution?

    (A) \(~~0=x^2+18x+145\)

    (B) \(~~0=x^2+18x-145\)

    (C) \(~~0=x^2-18x-145\)

    (D) \(~~0=x^2-18x+145\)

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    A polynomial with real coefficients has complex roots that come in pairs called complex conjugates. So, both 9+8i and 9-8i are solutions. From the two solutions, we can write the equation in factored form.

    \[0 ~=~ \big(x-(9+8i)\big)\big(x-(9-8i)\big)\]

    You can FOIL.

    \[0 ~=~ x^2-(9-8i)x-(9+8i)x+(9+8i)(9-8i)\]

    You can distribute the linear terms and FOIL the product of complex numbers.

    \[0~=~x^2-9x+8ix-9x-8ix+81-72i+72i-64i^2\]

    Remember, \(i^2=-1\).

    \[0~=~x^2-9x+8ix-9x-8ix+81-72i+72i+64\]

    Combine like terms.

    \[0=x^2-18x+145\]

    So the answer is D.

    
    
72. 
    

    Question

    prevnext

    Which of the following has \(6+3i\) as a solution?

    (A) \(~~0=x^2+12x+45\)

    (B) \(~~0=x^2+12x-45\)

    (C) \(~~0=x^2-12x+45\)

    (D) \(~~0=x^2-12x-45\)

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    A polynomial with real coefficients has complex roots that come in pairs called complex conjugates. So, both 6+3i and 6-3i are solutions. From the two solutions, we can write the equation in factored form.

    \[0 ~=~ \big(x-(6+3i)\big)\big(x-(6-3i)\big)\]

    You can FOIL.

    \[0 ~=~ x^2-(6-3i)x-(6+3i)x+(6+3i)(6-3i)\]

    You can distribute the linear terms and FOIL the product of complex numbers.

    \[0~=~x^2-6x+3ix-6x-3ix+36-18i+18i-9i^2\]

    Remember, \(i^2=-1\).

    \[0~=~x^2-6x+3ix-6x-3ix+36-18i+18i+9\]

    Combine like terms.

    \[0=x^2-12x+45\]

    So the answer is C.

    
    
73. 
    

    Question

    prevnext

    Which of the following has \(-1+8i\) as a solution?

    (A) \(~~0=x^2+2x-65\)

    (B) \(~~0=x^2+2x+65\)

    (C) \(~~0=x^2-2x+65\)

    (D) \(~~0=x^2-2x-65\)

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    A polynomial with real coefficients has complex roots that come in pairs called complex conjugates. So, both -1+8i and -1-8i are solutions. From the two solutions, we can write the equation in factored form.

    \[0 ~=~ \big(x-(-1+8i)\big)\big(x-(-1-8i)\big)\]

    You can FOIL.

    \[0 ~=~ x^2-(-1-8i)x-(-1+8i)x+(-1+8i)(-1-8i)\]

    You can distribute the linear terms and FOIL the product of complex numbers.

    \[0~=~x^2+x+8ix+x-8ix+1+8i-8i-64i^2\]

    Remember, \(i^2=-1\).

    \[0~=~x^2+x+8ix+x-8ix+1+8i-8i+64\]

    Combine like terms.

    \[0=x^2+2x+65\]

    So the answer is B.

    
    
74. 
    

    Question

    prevnext

    Which of the following has \(8-7i\) as a solution?

    (A) \(~~0=x^2+16x+113\)

    (B) \(~~0=x^2-16x+113\)

    (C) \(~~0=x^2-16x-113\)

    (D) \(~~0=x^2+16x-113\)

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    A polynomial with real coefficients has complex roots that come in pairs called complex conjugates. So, both 8-7i and 8+7i are solutions. From the two solutions, we can write the equation in factored form.

    \[0 ~=~ \big(x-(8-7i)\big)\big(x-(8+7i)\big)\]

    You can FOIL.

    \[0 ~=~ x^2-(8+7i)x-(8-7i)x+(8-7i)(8+7i)\]

    You can distribute the linear terms and FOIL the product of complex numbers.

    \[0~=~x^2-8x-7ix-8x+7ix+64+56i-56i-49i^2\]

    Remember, \(i^2=-1\).

    \[0~=~x^2-8x-7ix-8x+7ix+64+56i-56i+49\]

    Combine like terms.

    \[0=x^2-16x+113\]

    So the answer is B.

    
    
75. 
    

    Question

    prevnext

    Which of the following has \(1+i\) as a solution?

    (A) \(~~0=x^2-2x+2\)

    (B) \(~~0=x^2-2x-2\)

    (C) \(~~0=x^2+2x-2\)

    (D) \(~~0=x^2+2x+2\)

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    A polynomial with real coefficients has complex roots that come in pairs called complex conjugates. So, both 1+i and 1-i are solutions. From the two solutions, we can write the equation in factored form.

    \[0 ~=~ \big(x-(1+i)\big)\big(x-(1-i)\big)\]

    You can FOIL.

    \[0 ~=~ x^2-(1-i)x-(1+i)x+(1+i)(1-i)\]

    You can distribute the linear terms and FOIL the product of complex numbers.

    \[0~=~x^2-x+ix-x-ix+1-i+i-i^2\]

    Remember, \(i^2=-1\).

    \[0~=~x^2-x+ix-x-ix+1-i+i+1\]

    Combine like terms.

    \[0=x^2-2x+2\]

    So the answer is A.

    
    
76. 
    

    Question

    prevnext

    Which of the following has \(3+7i\) as a solution?

    (A) \(~~0=x^2+6x+58\)

    (B) \(~~0=x^2+6x-58\)

    (C) \(~~0=x^2-6x-58\)

    (D) \(~~0=x^2-6x+58\)

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    A polynomial with real coefficients has complex roots that come in pairs called complex conjugates. So, both 3+7i and 3-7i are solutions. From the two solutions, we can write the equation in factored form.

    \[0 ~=~ \big(x-(3+7i)\big)\big(x-(3-7i)\big)\]

    You can FOIL.

    \[0 ~=~ x^2-(3-7i)x-(3+7i)x+(3+7i)(3-7i)\]

    You can distribute the linear terms and FOIL the product of complex numbers.

    \[0~=~x^2-3x+7ix-3x-7ix+9-21i+21i-49i^2\]

    Remember, \(i^2=-1\).

    \[0~=~x^2-3x+7ix-3x-7ix+9-21i+21i+49\]

    Combine like terms.

    \[0=x^2-6x+58\]

    So the answer is D.

    
    
77. 
    

    Question

    prevnext

    Which of the following has \(-2-5i\) as a solution?

    (A) \(~~0=x^2-4x+29\)

    (B) \(~~0=x^2+4x+29\)

    (C) \(~~0=x^2-4x-29\)

    (D) \(~~0=x^2+4x-29\)

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    A polynomial with real coefficients has complex roots that come in pairs called complex conjugates. So, both -2-5i and -2+5i are solutions. From the two solutions, we can write the equation in factored form.

    \[0 ~=~ \big(x-(-2-5i)\big)\big(x-(-2+5i)\big)\]

    You can FOIL.

    \[0 ~=~ x^2-(-2+5i)x-(-2-5i)x+(-2-5i)(-2+5i)\]

    You can distribute the linear terms and FOIL the product of complex numbers.

    \[0~=~x^2+2x-5ix+2x+5ix+4-10i+10i-25i^2\]

    Remember, \(i^2=-1\).

    \[0~=~x^2+2x-5ix+2x+5ix+4-10i+10i+25\]

    Combine like terms.

    \[0=x^2+4x+29\]

    So the answer is B.

    
    
78. 
    

    Question

    prevnext

    Which of the following has \(1-3i\) as a solution?

    (A) \(~~0=x^2-2x-10\)

    (B) \(~~0=x^2-2x+10\)

    (C) \(~~0=x^2+2x-10\)

    (D) \(~~0=x^2+2x+10\)

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    A polynomial with real coefficients has complex roots that come in pairs called complex conjugates. So, both 1-3i and 1+3i are solutions. From the two solutions, we can write the equation in factored form.

    \[0 ~=~ \big(x-(1-3i)\big)\big(x-(1+3i)\big)\]

    You can FOIL.

    \[0 ~=~ x^2-(1+3i)x-(1-3i)x+(1-3i)(1+3i)\]

    You can distribute the linear terms and FOIL the product of complex numbers.

    \[0~=~x^2-x-3ix-x+3ix+1+3i-3i-9i^2\]

    Remember, \(i^2=-1\).

    \[0~=~x^2-x-3ix-x+3ix+1+3i-3i+9\]

    Combine like terms.

    \[0=x^2-2x+10\]

    So the answer is B.

    
    
79. 
    

    Question

    prevnext

    Which of the following has \(5+4i\) as a solution?

    (A) \(~~0=x^2+10x-41\)

    (B) \(~~0=x^2-10x+41\)

    (C) \(~~0=x^2+10x+41\)

    (D) \(~~0=x^2-10x-41\)

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    A polynomial with real coefficients has complex roots that come in pairs called complex conjugates. So, both 5+4i and 5-4i are solutions. From the two solutions, we can write the equation in factored form.

    \[0 ~=~ \big(x-(5+4i)\big)\big(x-(5-4i)\big)\]

    You can FOIL.

    \[0 ~=~ x^2-(5-4i)x-(5+4i)x+(5+4i)(5-4i)\]

    You can distribute the linear terms and FOIL the product of complex numbers.

    \[0~=~x^2-5x+4ix-5x-4ix+25-20i+20i-16i^2\]

    Remember, \(i^2=-1\).

    \[0~=~x^2-5x+4ix-5x-4ix+25-20i+20i+16\]

    Combine like terms.

    \[0=x^2-10x+41\]

    So the answer is B.

    
    
80. 
    

    Question

    prevnext

    Which of the following has \(-3-i\) as a solution?

    (A) \(~~0=x^2+6x+10\)

    (B) \(~~0=x^2-6x+10\)

    (C) \(~~0=x^2-6x-10\)

    (D) \(~~0=x^2+6x-10\)

    
    
    1. A
    2. B
    3. C
    4. D
    
    

    Solution

    Show solution

    A polynomial with real coefficients has complex roots that come in pairs called complex conjugates. So, both -3-i and -3+i are solutions. From the two solutions, we can write the equation in factored form.

    \[0 ~=~ \big(x-(-3-i)\big)\big(x-(-3+i)\big)\]

    You can FOIL.

    \[0 ~=~ x^2-(-3+i)x-(-3-i)x+(-3-i)(-3+i)\]

    You can distribute the linear terms and FOIL the product of complex numbers.

    \[0~=~x^2+3x-ix+3x+ix+9-3i+3i-i^2\]

    Remember, \(i^2=-1\).

    \[0~=~x^2+3x-ix+3x+ix+9-3i+3i+1\]

    Combine like terms.

    \[0=x^2+6x+10\]

    So the answer is A.

    
    
81. 
    

    Question

    prevnext

    Simplify the quotient of complex numbers by rationalizing the denominator.

    \[\frac{2-i}{6+3i}\]

    Your answer (when simplified) will have the form \[\frac{a+bi}{c}\] If \(c>0\), what are the values of \(a\), \(b\), and \(c\)?

    • \(a=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(b=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(c=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    
    
    
    

    Solution

    Show solution

    Multiply the numerator and denominator by the conjugate of the denominator.

    \[\frac{2-i}{6+3i} \cdot \frac{6-3i}{6-3i}\]

    FOIL the numerator and denominator.

    \[\frac{(2)(6)+(2)(-3)i+(-1)(6)i+(-1)(-3)i^2}{(6)(6)+(6)(-3)i+(3)(6)i+(3)(-3)i^2}\]

    Remember, \(i^2=-1\).

    \[\frac{(12)+(-6)i+(-6)i+(3)(-1)}{(36)+(-18)i+(18)i+(-9)(-1)} \]

    Combine like terms.

    \[\frac{9-12i}{45+0i}\]

    The denominator is now real, because \(0i=0\).

    \[\frac{9-12i}{45}\]

    The greatest common factor of 9 and -12 and 45 is 3. So, you can divide the numerator and denominator by 3.

    \[\frac{3-4i}{15}\]

    Thus, \(a=3\) and \(b=-4\) and \(c=15\).

    
    
82. 
    

    Question

    prevnext

    Simplify the quotient of complex numbers by rationalizing the denominator.

    \[\frac{5-7i}{6+3i}\]

    Your answer (when simplified) will have the form \[\frac{a+bi}{c}\] If \(c>0\), what are the values of \(a\), \(b\), and \(c\)?

    • \(a=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(b=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(c=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    
    
    
    

    Solution

    Show solution

    Multiply the numerator and denominator by the conjugate of the denominator.

    \[\frac{5-7i}{6+3i} \cdot \frac{6-3i}{6-3i}\]

    FOIL the numerator and denominator.

    \[\frac{(5)(6)+(5)(-3)i+(-7)(6)i+(-7)(-3)i^2}{(6)(6)+(6)(-3)i+(3)(6)i+(3)(-3)i^2}\]

    Remember, \(i^2=-1\).

    \[\frac{(30)+(-15)i+(-42)i+(21)(-1)}{(36)+(-18)i+(18)i+(-9)(-1)} \]

    Combine like terms.

    \[\frac{9-57i}{45+0i}\]

    The denominator is now real, because \(0i=0\).

    \[\frac{9-57i}{45}\]

    The greatest common factor of 9 and -57 and 45 is 3. So, you can divide the numerator and denominator by 3.

    \[\frac{3-19i}{15}\]

    Thus, \(a=3\) and \(b=-19\) and \(c=15\).

    
    
83. 
    

    Question

    prevnext

    Simplify the quotient of complex numbers by rationalizing the denominator.

    \[\frac{-8-3i}{-1+2i}\]

    Your answer (when simplified) will have the form \[\frac{a+bi}{c}\] If \(c>0\), what are the values of \(a\), \(b\), and \(c\)?

    • \(a=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(b=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(c=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    
    
    
    

    Solution

    Show solution

    Multiply the numerator and denominator by the conjugate of the denominator.

    \[\frac{-8-3i}{-1+2i} \cdot \frac{-1-2i}{-1-2i}\]

    FOIL the numerator and denominator.

    \[\frac{(-8)(-1)+(-8)(-2)i+(-3)(-1)i+(-3)(-2)i^2}{(-1)(-1)+(-1)(-2)i+(2)(-1)i+(2)(-2)i^2}\]

    Remember, \(i^2=-1\).

    \[\frac{(8)+(16)i+(3)i+(6)(-1)}{(1)+(2)i+(-2)i+(-4)(-1)} \]

    Combine like terms.

    \[\frac{2+19i}{5+0i}\]

    The denominator is now real, because \(0i=0\).

    \[\frac{2+19i}{5}\]

    The greatest common factor of 2 and 19 and 5 is 1. So, you can divide the numerator and denominator by 1.

    \[\frac{2+19i}{5}\]

    Thus, \(a=2\) and \(b=19\) and \(c=5\).

    
    
84. 
    

    Question

    prevnext

    Simplify the quotient of complex numbers by rationalizing the denominator.

    \[\frac{-4-7i}{9+2i}\]

    Your answer (when simplified) will have the form \[\frac{a+bi}{c}\] If \(c>0\), what are the values of \(a\), \(b\), and \(c\)?

    • \(a=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(b=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(c=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    
    
    
    

    Solution

    Show solution

    Multiply the numerator and denominator by the conjugate of the denominator.

    \[\frac{-4-7i}{9+2i} \cdot \frac{9-2i}{9-2i}\]

    FOIL the numerator and denominator.

    \[\frac{(-4)(9)+(-4)(-2)i+(-7)(9)i+(-7)(-2)i^2}{(9)(9)+(9)(-2)i+(2)(9)i+(2)(-2)i^2}\]

    Remember, \(i^2=-1\).

    \[\frac{(-36)+(8)i+(-63)i+(14)(-1)}{(81)+(-18)i+(18)i+(-4)(-1)} \]

    Combine like terms.

    \[\frac{-50-55i}{85+0i}\]

    The denominator is now real, because \(0i=0\).

    \[\frac{-50-55i}{85}\]

    The greatest common factor of -50 and -55 and 85 is 5. So, you can divide the numerator and denominator by 5.

    \[\frac{-10-11i}{17}\]

    Thus, \(a=-10\) and \(b=-11\) and \(c=17\).

    
    
85. 
    

    Question

    prevnext

    Simplify the quotient of complex numbers by rationalizing the denominator.

    \[\frac{1-5i}{2+4i}\]

    Your answer (when simplified) will have the form \[\frac{a+bi}{c}\] If \(c>0\), what are the values of \(a\), \(b\), and \(c\)?

    • \(a=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(b=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(c=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    
    
    
    

    Solution

    Show solution

    Multiply the numerator and denominator by the conjugate of the denominator.

    \[\frac{1-5i}{2+4i} \cdot \frac{2-4i}{2-4i}\]

    FOIL the numerator and denominator.

    \[\frac{(1)(2)+(1)(-4)i+(-5)(2)i+(-5)(-4)i^2}{(2)(2)+(2)(-4)i+(4)(2)i+(4)(-4)i^2}\]

    Remember, \(i^2=-1\).

    \[\frac{(2)+(-4)i+(-10)i+(20)(-1)}{(4)+(-8)i+(8)i+(-16)(-1)} \]

    Combine like terms.

    \[\frac{-18-14i}{20+0i}\]

    The denominator is now real, because \(0i=0\).

    \[\frac{-18-14i}{20}\]

    The greatest common factor of -18 and -14 and 20 is 2. So, you can divide the numerator and denominator by 2.

    \[\frac{-9-7i}{10}\]

    Thus, \(a=-9\) and \(b=-7\) and \(c=10\).

    
    
86. 
    

    Question

    prevnext

    Simplify the quotient of complex numbers by rationalizing the denominator.

    \[\frac{6-4i}{3+i}\]

    Your answer (when simplified) will have the form \[\frac{a+bi}{c}\] If \(c>0\), what are the values of \(a\), \(b\), and \(c\)?

    • \(a=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(b=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(c=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    
    
    
    

    Solution

    Show solution

    Multiply the numerator and denominator by the conjugate of the denominator.

    \[\frac{6-4i}{3+i} \cdot \frac{3-i}{3-i}\]

    FOIL the numerator and denominator.

    \[\frac{(6)(3)+(6)(-1)i+(-4)(3)i+(-4)(-1)i^2}{(3)(3)+(3)(-1)i+(1)(3)i+(1)(-1)i^2}\]

    Remember, \(i^2=-1\).

    \[\frac{(18)+(-6)i+(-12)i+(4)(-1)}{(9)+(-3)i+(3)i+(-1)(-1)} \]

    Combine like terms.

    \[\frac{14-18i}{10+0i}\]

    The denominator is now real, because \(0i=0\).

    \[\frac{14-18i}{10}\]

    The greatest common factor of 14 and -18 and 10 is 2. So, you can divide the numerator and denominator by 2.

    \[\frac{7-9i}{5}\]

    Thus, \(a=7\) and \(b=-9\) and \(c=5\).

    
    
87. 
    

    Question

    prevnext

    Simplify the quotient of complex numbers by rationalizing the denominator.

    \[\frac{5+7i}{2+6i}\]

    Your answer (when simplified) will have the form \[\frac{a+bi}{c}\] If \(c>0\), what are the values of \(a\), \(b\), and \(c\)?

    • \(a=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(b=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(c=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    
    
    
    

    Solution

    Show solution

    Multiply the numerator and denominator by the conjugate of the denominator.

    \[\frac{5+7i}{2+6i} \cdot \frac{2-6i}{2-6i}\]

    FOIL the numerator and denominator.

    \[\frac{(5)(2)+(5)(-6)i+(7)(2)i+(7)(-6)i^2}{(2)(2)+(2)(-6)i+(6)(2)i+(6)(-6)i^2}\]

    Remember, \(i^2=-1\).

    \[\frac{(10)+(-30)i+(14)i+(-42)(-1)}{(4)+(-12)i+(12)i+(-36)(-1)} \]

    Combine like terms.

    \[\frac{52-16i}{40+0i}\]

    The denominator is now real, because \(0i=0\).

    \[\frac{52-16i}{40}\]

    The greatest common factor of 52 and -16 and 40 is 4. So, you can divide the numerator and denominator by 4.

    \[\frac{13-4i}{10}\]

    Thus, \(a=13\) and \(b=-4\) and \(c=10\).

    
    
88. 
    

    Question

    prevnext

    Simplify the quotient of complex numbers by rationalizing the denominator.

    \[\frac{-7-4i}{-3+i}\]

    Your answer (when simplified) will have the form \[\frac{a+bi}{c}\] If \(c>0\), what are the values of \(a\), \(b\), and \(c\)?

    • \(a=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(b=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(c=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    
    
    
    

    Solution

    Show solution

    Multiply the numerator and denominator by the conjugate of the denominator.

    \[\frac{-7-4i}{-3+i} \cdot \frac{-3-i}{-3-i}\]

    FOIL the numerator and denominator.

    \[\frac{(-7)(-3)+(-7)(-1)i+(-4)(-3)i+(-4)(-1)i^2}{(-3)(-3)+(-3)(-1)i+(1)(-3)i+(1)(-1)i^2}\]

    Remember, \(i^2=-1\).

    \[\frac{(21)+(7)i+(12)i+(4)(-1)}{(9)+(3)i+(-3)i+(-1)(-1)} \]

    Combine like terms.

    \[\frac{17+19i}{10+0i}\]

    The denominator is now real, because \(0i=0\).

    \[\frac{17+19i}{10}\]

    The greatest common factor of 17 and 19 and 10 is 1. So, you can divide the numerator and denominator by 1.

    \[\frac{17+19i}{10}\]

    Thus, \(a=17\) and \(b=19\) and \(c=10\).

    
    
89. 
    

    Question

    prevnext

    Simplify the quotient of complex numbers by rationalizing the denominator.

    \[\frac{1+9i}{-2+6i}\]

    Your answer (when simplified) will have the form \[\frac{a+bi}{c}\] If \(c>0\), what are the values of \(a\), \(b\), and \(c\)?

    • \(a=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(b=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(c=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    
    
    
    

    Solution

    Show solution

    Multiply the numerator and denominator by the conjugate of the denominator.

    \[\frac{1+9i}{-2+6i} \cdot \frac{-2-6i}{-2-6i}\]

    FOIL the numerator and denominator.

    \[\frac{(1)(-2)+(1)(-6)i+(9)(-2)i+(9)(-6)i^2}{(-2)(-2)+(-2)(-6)i+(6)(-2)i+(6)(-6)i^2}\]

    Remember, \(i^2=-1\).

    \[\frac{(-2)+(-6)i+(-18)i+(-54)(-1)}{(4)+(12)i+(-12)i+(-36)(-1)} \]

    Combine like terms.

    \[\frac{52-24i}{40+0i}\]

    The denominator is now real, because \(0i=0\).

    \[\frac{52-24i}{40}\]

    The greatest common factor of 52 and -24 and 40 is 4. So, you can divide the numerator and denominator by 4.

    \[\frac{13-6i}{10}\]

    Thus, \(a=13\) and \(b=-6\) and \(c=10\).

    
    
90. 
    

    Question

    prevnext

    Simplify the quotient of complex numbers by rationalizing the denominator.

    \[\frac{-8+4i}{9+3i}\]

    Your answer (when simplified) will have the form \[\frac{a+bi}{c}\] If \(c>0\), what are the values of \(a\), \(b\), and \(c\)?

    • \(a=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(b=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    • \(c=\) -------- -20-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-101234567891011121314151617181920
    
    
    
    

    Solution

    Show solution

    Multiply the numerator and denominator by the conjugate of the denominator.

    \[\frac{-8+4i}{9+3i} \cdot \frac{9-3i}{9-3i}\]

    FOIL the numerator and denominator.

    \[\frac{(-8)(9)+(-8)(-3)i+(4)(9)i+(4)(-3)i^2}{(9)(9)+(9)(-3)i+(3)(9)i+(3)(-3)i^2}\]

    Remember, \(i^2=-1\).

    \[\frac{(-72)+(24)i+(36)i+(-12)(-1)}{(81)+(-27)i+(27)i+(-9)(-1)} \]

    Combine like terms.

    \[\frac{-60+60i}{90+0i}\]

    The denominator is now real, because \(0i=0\).

    \[\frac{-60+60i}{90}\]

    The greatest common factor of -60 and 60 and 90 is 30. So, you can divide the numerator and denominator by 30.

    \[\frac{-2+2i}{3}\]

    Thus, \(a=-2\) and \(b=2\) and \(c=3\).

    
    
91. 
    

    Question

    prevnext

    In Chemistry, the pH is used to measure the acidity of a solution. The pH is calculated from the concentration of hydrogen ions (in moles per liter).

    \[\mathrm{pH} = -\log_{10}\big(\left[\mathrm{H^{+}}\right]\big)\]

    The table below lists the (rounded) base-10 logarithm of integers between 1 and 9.

    \(x\)	\(\log_{10}(x)\)
    1	0.000
    2	0.301
    3	0.477
    4	0.602
    5	0.699
    6	0.778
    7	0.845
    8	0.903
    9	0.954

    If the concentration of hydrogen ions is \(\left[\mathrm{H^{+}}\right]=8\cdot 10^{-11}\) moles per liter, what is the pH? Please use the table, and do not round your answer any more than the table already has.

    
    

    Solution

    Show solution

    Plug that concentration into the pH formula.

    \[\mathrm{pH} = -\log_{10}\left(8\cdot 10^{-11}\right)\]

    Remember that a log of a product equals a sum of logs.

    \[\mathrm{pH} = -\left(\log_{10}(8)+\log_{10}\left(10^{-11}\right)\right)\]

    Use that table. Also, \(\log_{10}(10^p)\equiv p\) for all \(p\). So,

    \[\mathrm{pH} = -\left(0.903+(-11)\right)\]

    Distribute the negative.

    \[\mathrm{pH} = -0.903+11\]

    Evaluate \(11-0.903\).

    \[\mathrm{pH} = 10.097\]

    
    
92. 
    

    Question

    prevnext

    In Chemistry, the pH is used to measure the acidity of a solution. The pH is calculated from the concentration of hydrogen ions (in moles per liter).

    \[\mathrm{pH} = -\log_{10}\big(\left[\mathrm{H^{+}}\right]\big)\]

    The table below lists the (rounded) base-10 logarithm of integers between 1 and 9.

    \(x\)	\(\log_{10}(x)\)
    1	0.000
    2	0.301
    3	0.477
    4	0.602
    5	0.699
    6	0.778
    7	0.845
    8	0.903
    9	0.954

    If the concentration of hydrogen ions is \(\left[\mathrm{H^{+}}\right]=5\cdot 10^{-1}\) moles per liter, what is the pH? Please use the table, and do not round your answer any more than the table already has.

    
    

    Solution

    Show solution

    Plug that concentration into the pH formula.

    \[\mathrm{pH} = -\log_{10}\left(5\cdot 10^{-1}\right)\]

    Remember that a log of a product equals a sum of logs.

    \[\mathrm{pH} = -\left(\log_{10}(5)+\log_{10}\left(10^{-1}\right)\right)\]

    Use that table. Also, \(\log_{10}(10^p)\equiv p\) for all \(p\). So,

    \[\mathrm{pH} = -\left(0.699+(-1)\right)\]

    Distribute the negative.

    \[\mathrm{pH} = -0.699+1\]

    Evaluate \(1-0.699\).

    \[\mathrm{pH} = 0.301\]

    
    
93. 
    

    Question

    prevnext

    In Chemistry, the pH is used to measure the acidity of a solution. The pH is calculated from the concentration of hydrogen ions (in moles per liter).

    \[\mathrm{pH} = -\log_{10}\big(\left[\mathrm{H^{+}}\right]\big)\]

    The table below lists the (rounded) base-10 logarithm of integers between 1 and 9.

    \(x\)	\(\log_{10}(x)\)
    1	0.000
    2	0.301
    3	0.477
    4	0.602
    5	0.699
    6	0.778
    7	0.845
    8	0.903
    9	0.954

    If the concentration of hydrogen ions is \(\left[\mathrm{H^{+}}\right]=9\cdot 10^{-1}\) moles per liter, what is the pH? Please use the table, and do not round your answer any more than the table already has.

    
    

    Solution

    Show solution

    Plug that concentration into the pH formula.

    \[\mathrm{pH} = -\log_{10}\left(9\cdot 10^{-1}\right)\]

    Remember that a log of a product equals a sum of logs.

    \[\mathrm{pH} = -\left(\log_{10}(9)+\log_{10}\left(10^{-1}\right)\right)\]

    Use that table. Also, \(\log_{10}(10^p)\equiv p\) for all \(p\). So,

    \[\mathrm{pH} = -\left(0.954+(-1)\right)\]

    Distribute the negative.

    \[\mathrm{pH} = -0.954+1\]

    Evaluate \(1-0.954\).

    \[\mathrm{pH} = 0.046\]

    
    
94. 
    

    Question

    prevnext

    In Chemistry, the pH is used to measure the acidity of a solution. The pH is calculated from the concentration of hydrogen ions (in moles per liter).

    \[\mathrm{pH} = -\log_{10}\big(\left[\mathrm{H^{+}}\right]\big)\]

    The table below lists the (rounded) base-10 logarithm of integers between 1 and 9.

    \(x\)	\(\log_{10}(x)\)
    1	0.000
    2	0.301
    3	0.477
    4	0.602
    5	0.699
    6	0.778
    7	0.845
    8	0.903
    9	0.954

    If the concentration of hydrogen ions is \(\left[\mathrm{H^{+}}\right]=5\cdot 10^{-2}\) moles per liter, what is the pH? Please use the table, and do not round your answer any more than the table already has.

    
    

    Solution

    Show solution

    Plug that concentration into the pH formula.

    \[\mathrm{pH} = -\log_{10}\left(5\cdot 10^{-2}\right)\]

    Remember that a log of a product equals a sum of logs.

    \[\mathrm{pH} = -\left(\log_{10}(5)+\log_{10}\left(10^{-2}\right)\right)\]

    Use that table. Also, \(\log_{10}(10^p)\equiv p\) for all \(p\). So,

    \[\mathrm{pH} = -\left(0.699+(-2)\right)\]

    Distribute the negative.

    \[\mathrm{pH} = -0.699+2\]

    Evaluate \(2-0.699\).

    \[\mathrm{pH} = 1.301\]

    
    
95. 
    

    Question

    prevnext

    In Chemistry, the pH is used to measure the acidity of a solution. The pH is calculated from the concentration of hydrogen ions (in moles per liter).

    \[\mathrm{pH} = -\log_{10}\big(\left[\mathrm{H^{+}}\right]\big)\]

    The table below lists the (rounded) base-10 logarithm of integers between 1 and 9.

    \(x\)	\(\log_{10}(x)\)
    1	0.000
    2	0.301
    3	0.477
    4	0.602
    5	0.699
    6	0.778
    7	0.845
    8	0.903
    9	0.954

    If the concentration of hydrogen ions is \(\left[\mathrm{H^{+}}\right]=4\cdot 10^{-6}\) moles per liter, what is the pH? Please use the table, and do not round your answer any more than the table already has.

    
    

    Solution

    Show solution

    Plug that concentration into the pH formula.

    \[\mathrm{pH} = -\log_{10}\left(4\cdot 10^{-6}\right)\]

    Remember that a log of a product equals a sum of logs.

    \[\mathrm{pH} = -\left(\log_{10}(4)+\log_{10}\left(10^{-6}\right)\right)\]

    Use that table. Also, \(\log_{10}(10^p)\equiv p\) for all \(p\). So,

    \[\mathrm{pH} = -\left(0.602+(-6)\right)\]

    Distribute the negative.

    \[\mathrm{pH} = -0.602+6\]

    Evaluate \(6-0.602\).

    \[\mathrm{pH} = 5.398\]

    
    
96. 
    

    Question

    prevnext

    In Chemistry, the pH is used to measure the acidity of a solution. The pH is calculated from the concentration of hydrogen ions (in moles per liter).

    \[\mathrm{pH} = -\log_{10}\big(\left[\mathrm{H^{+}}\right]\big)\]

    The table below lists the (rounded) base-10 logarithm of integers between 1 and 9.

    \(x\)	\(\log_{10}(x)\)
    1	0.000
    2	0.301
    3	0.477
    4	0.602
    5	0.699
    6	0.778
    7	0.845
    8	0.903
    9	0.954

    If the concentration of hydrogen ions is \(\left[\mathrm{H^{+}}\right]=5\cdot 10^{-12}\) moles per liter, what is the pH? Please use the table, and do not round your answer any more than the table already has.

    
    

    Solution

    Show solution

    Plug that concentration into the pH formula.

    \[\mathrm{pH} = -\log_{10}\left(5\cdot 10^{-12}\right)\]

    Remember that a log of a product equals a sum of logs.

    \[\mathrm{pH} = -\left(\log_{10}(5)+\log_{10}\left(10^{-12}\right)\right)\]

    Use that table. Also, \(\log_{10}(10^p)\equiv p\) for all \(p\). So,

    \[\mathrm{pH} = -\left(0.699+(-12)\right)\]

    Distribute the negative.

    \[\mathrm{pH} = -0.699+12\]

    Evaluate \(12-0.699\).

    \[\mathrm{pH} = 11.301\]

    
    
97. 
    

    Question

    prevnext

    In Chemistry, the pH is used to measure the acidity of a solution. The pH is calculated from the concentration of hydrogen ions (in moles per liter).

    \[\mathrm{pH} = -\log_{10}\big(\left[\mathrm{H^{+}}\right]\big)\]

    The table below lists the (rounded) base-10 logarithm of integers between 1 and 9.

    \(x\)	\(\log_{10}(x)\)
    1	0.000
    2	0.301
    3	0.477
    4	0.602
    5	0.699
    6	0.778
    7	0.845
    8	0.903
    9	0.954

    If the concentration of hydrogen ions is \(\left[\mathrm{H^{+}}\right]=9\cdot 10^{-9}\) moles per liter, what is the pH? Please use the table, and do not round your answer any more than the table already has.

    
    

    Solution

    Show solution

    Plug that concentration into the pH formula.

    \[\mathrm{pH} = -\log_{10}\left(9\cdot 10^{-9}\right)\]

    Remember that a log of a product equals a sum of logs.

    \[\mathrm{pH} = -\left(\log_{10}(9)+\log_{10}\left(10^{-9}\right)\right)\]

    Use that table. Also, \(\log_{10}(10^p)\equiv p\) for all \(p\). So,

    \[\mathrm{pH} = -\left(0.954+(-9)\right)\]

    Distribute the negative.

    \[\mathrm{pH} = -0.954+9\]

    Evaluate \(9-0.954\).

    \[\mathrm{pH} = 8.046\]

    
    
98. 
    

    Question

    prevnext

    In Chemistry, the pH is used to measure the acidity of a solution. The pH is calculated from the concentration of hydrogen ions (in moles per liter).

    \[\mathrm{pH} = -\log_{10}\big(\left[\mathrm{H^{+}}\right]\big)\]

    The table below lists the (rounded) base-10 logarithm of integers between 1 and 9.

    \(x\)	\(\log_{10}(x)\)
    1	0.000
    2	0.301
    3	0.477
    4	0.602
    5	0.699
    6	0.778
    7	0.845
    8	0.903
    9	0.954

    If the concentration of hydrogen ions is \(\left[\mathrm{H^{+}}\right]=4\cdot 10^{-2}\) moles per liter, what is the pH? Please use the table, and do not round your answer any more than the table already has.

    
    

    Solution

    Show solution

    Plug that concentration into the pH formula.

    \[\mathrm{pH} = -\log_{10}\left(4\cdot 10^{-2}\right)\]

    Remember that a log of a product equals a sum of logs.

    \[\mathrm{pH} = -\left(\log_{10}(4)+\log_{10}\left(10^{-2}\right)\right)\]

    Use that table. Also, \(\log_{10}(10^p)\equiv p\) for all \(p\). So,

    \[\mathrm{pH} = -\left(0.602+(-2)\right)\]

    Distribute the negative.

    \[\mathrm{pH} = -0.602+2\]

    Evaluate \(2-0.602\).

    \[\mathrm{pH} = 1.398\]

    
    
99. 
    

    Question

    prevnext

    In Chemistry, the pH is used to measure the acidity of a solution. The pH is calculated from the concentration of hydrogen ions (in moles per liter).

    \[\mathrm{pH} = -\log_{10}\big(\left[\mathrm{H^{+}}\right]\big)\]

    The table below lists the (rounded) base-10 logarithm of integers between 1 and 9.

    \(x\)	\(\log_{10}(x)\)
    1	0.000
    2	0.301
    3	0.477
    4	0.602
    5	0.699
    6	0.778
    7	0.845
    8	0.903
    9	0.954

    If the concentration of hydrogen ions is \(\left[\mathrm{H^{+}}\right]=4\cdot 10^{-8}\) moles per liter, what is the pH? Please use the table, and do not round your answer any more than the table already has.

    
    

    Solution

    Show solution

    Plug that concentration into the pH formula.

    \[\mathrm{pH} = -\log_{10}\left(4\cdot 10^{-8}\right)\]

    Remember that a log of a product equals a sum of logs.

    \[\mathrm{pH} = -\left(\log_{10}(4)+\log_{10}\left(10^{-8}\right)\right)\]

    Use that table. Also, \(\log_{10}(10^p)\equiv p\) for all \(p\). So,

    \[\mathrm{pH} = -\left(0.602+(-8)\right)\]

    Distribute the negative.

    \[\mathrm{pH} = -0.602+8\]

    Evaluate \(8-0.602\).

    \[\mathrm{pH} = 7.398\]

    
    
100. 
     

     Question

     prevnext

     In Chemistry, the pH is used to measure the acidity of a solution. The pH is calculated from the concentration of hydrogen ions (in moles per liter).

     \[\mathrm{pH} = -\log_{10}\big(\left[\mathrm{H^{+}}\right]\big)\]

     The table below lists the (rounded) base-10 logarithm of integers between 1 and 9.

     \(x\)	\(\log_{10}(x)\)
     1	0.000
     2	0.301
     3	0.477
     4	0.602
     5	0.699
     6	0.778
     7	0.845
     8	0.903
     9	0.954

     If the concentration of hydrogen ions is \(\left[\mathrm{H^{+}}\right]=9\cdot 10^{-13}\) moles per liter, what is the pH? Please use the table, and do not round your answer any more than the table already has.

     
     

     Solution

     Show solution

     Plug that concentration into the pH formula.

     \[\mathrm{pH} = -\log_{10}\left(9\cdot 10^{-13}\right)\]

     Remember that a log of a product equals a sum of logs.

     \[\mathrm{pH} = -\left(\log_{10}(9)+\log_{10}\left(10^{-13}\right)\right)\]

     Use that table. Also, \(\log_{10}(10^p)\equiv p\) for all \(p\). So,

     \[\mathrm{pH} = -\left(0.954+(-13)\right)\]

     Distribute the negative.

     \[\mathrm{pH} = -0.954+13\]

     Evaluate \(13-0.954\).

     \[\mathrm{pH} = 12.046\]

     
     
101. 
     

     Question

     prevnext

     You can verify the exponential relationship below. \[4^{3} = 64\]

     These same three numbers can be used to satisfy the following logarithmic equation.

     \[\log_{a}(b) = c\]

     Determine values for \(a\), \(b\), and \(c\).

     • \(a =\) -------- 3464
     • \(b =\) -------- 3464
     • \(c =\) -------- 3464
     
     
     
     

     Solution

     Show solution

     Personally, I prefer thinking this through algebraically. We can take a log of both sides.

     \[\log\left(4^{3}\right) = \log(64)\]

     When the argument of a log function has an exponent, that exponent can be brought down as a factor outside the log function.

     \[3\cdot\log(4) = \log(64)\]

     We can divide both sides by \(\log(4)\). \[3 = \frac{\log(64)}{\log(4)}\]

     Apply the change of base rule. \[3 = \log_{4}(64)\] Switch the sides.

     \[\log_{4}(64) = 3\]

     So, \(a=4\) and \(b=64\) and \(c=3\).

     
     
102. 
     

     Question

     prevnext

     You can verify the exponential relationship below. \[6^{3} = 216\]

     These same three numbers can be used to satisfy the following logarithmic equation.

     \[\log_{a}(b) = c\]

     Determine values for \(a\), \(b\), and \(c\).

     • \(a =\) -------- 36216
     • \(b =\) -------- 36216
     • \(c =\) -------- 36216
     
     
     
     

     Solution

     Show solution

     Personally, I prefer thinking this through algebraically. We can take a log of both sides.

     \[\log\left(6^{3}\right) = \log(216)\]

     When the argument of a log function has an exponent, that exponent can be brought down as a factor outside the log function.

     \[3\cdot\log(6) = \log(216)\]

     We can divide both sides by \(\log(6)\). \[3 = \frac{\log(216)}{\log(6)}\]

     Apply the change of base rule. \[3 = \log_{6}(216)\] Switch the sides.

     \[\log_{6}(216) = 3\]

     So, \(a=6\) and \(b=216\) and \(c=3\).

     
     
103. 
     

     Question

     prevnext

     You can verify the exponential relationship below. \[3^{8} = 6561\]

     These same three numbers can be used to satisfy the following logarithmic equation.

     \[\log_{a}(b) = c\]

     Determine values for \(a\), \(b\), and \(c\).

     • \(a =\) -------- 386561
     • \(b =\) -------- 386561
     • \(c =\) -------- 386561
     
     
     
     

     Solution

     Show solution

     Personally, I prefer thinking this through algebraically. We can take a log of both sides.

     \[\log\left(3^{8}\right) = \log(6561)\]

     When the argument of a log function has an exponent, that exponent can be brought down as a factor outside the log function.

     \[8\cdot\log(3) = \log(6561)\]

     We can divide both sides by \(\log(3)\). \[8 = \frac{\log(6561)}{\log(3)}\]

     Apply the change of base rule. \[8 = \log_{3}(6561)\] Switch the sides.

     \[\log_{3}(6561) = 8\]

     So, \(a=3\) and \(b=6561\) and \(c=8\).

     
     
104. 
     

     Question

     prevnext

     You can verify the exponential relationship below. \[7^{3} = 343\]

     These same three numbers can be used to satisfy the following logarithmic equation.

     \[\log_{a}(b) = c\]

     Determine values for \(a\), \(b\), and \(c\).

     • \(a =\) -------- 37343
     • \(b =\) -------- 37343
     • \(c =\) -------- 37343
     
     
     
     

     Solution

     Show solution

     Personally, I prefer thinking this through algebraically. We can take a log of both sides.

     \[\log\left(7^{3}\right) = \log(343)\]

     When the argument of a log function has an exponent, that exponent can be brought down as a factor outside the log function.

     \[3\cdot\log(7) = \log(343)\]

     We can divide both sides by \(\log(7)\). \[3 = \frac{\log(343)}{\log(7)}\]

     Apply the change of base rule. \[3 = \log_{7}(343)\] Switch the sides.

     \[\log_{7}(343) = 3\]

     So, \(a=7\) and \(b=343\) and \(c=3\).

     
     
105. 
     

     Question

     prevnext

     You can verify the exponential relationship below. \[6^{2} = 36\]

     These same three numbers can be used to satisfy the following logarithmic equation.

     \[\log_{a}(b) = c\]

     Determine values for \(a\), \(b\), and \(c\).

     • \(a =\) -------- 2636
     • \(b =\) -------- 2636
     • \(c =\) -------- 2636
     
     
     
     

     Solution

     Show solution

     Personally, I prefer thinking this through algebraically. We can take a log of both sides.

     \[\log\left(6^{2}\right) = \log(36)\]

     When the argument of a log function has an exponent, that exponent can be brought down as a factor outside the log function.

     \[2\cdot\log(6) = \log(36)\]

     We can divide both sides by \(\log(6)\). \[2 = \frac{\log(36)}{\log(6)}\]

     Apply the change of base rule. \[2 = \log_{6}(36)\] Switch the sides.

     \[\log_{6}(36) = 2\]

     So, \(a=6\) and \(b=36\) and \(c=2\).

     
     
106. 
     

     Question

     prevnext

     You can verify the exponential relationship below. \[8^{2} = 64\]

     These same three numbers can be used to satisfy the following logarithmic equation.

     \[\log_{a}(b) = c\]

     Determine values for \(a\), \(b\), and \(c\).

     • \(a =\) -------- 2864
     • \(b =\) -------- 2864
     • \(c =\) -------- 2864
     
     
     
     

     Solution

     Show solution

     Personally, I prefer thinking this through algebraically. We can take a log of both sides.

     \[\log\left(8^{2}\right) = \log(64)\]

     When the argument of a log function has an exponent, that exponent can be brought down as a factor outside the log function.

     \[2\cdot\log(8) = \log(64)\]

     We can divide both sides by \(\log(8)\). \[2 = \frac{\log(64)}{\log(8)}\]

     Apply the change of base rule. \[2 = \log_{8}(64)\] Switch the sides.

     \[\log_{8}(64) = 2\]

     So, \(a=8\) and \(b=64\) and \(c=2\).

     
     
107. 
     

     Question

     prevnext

     You can verify the exponential relationship below. \[9^{4} = 6561\]

     These same three numbers can be used to satisfy the following logarithmic equation.

     \[\log_{a}(b) = c\]

     Determine values for \(a\), \(b\), and \(c\).

     • \(a =\) -------- 496561
     • \(b =\) -------- 496561
     • \(c =\) -------- 496561
     
     
     
     

     Solution

     Show solution

     Personally, I prefer thinking this through algebraically. We can take a log of both sides.

     \[\log\left(9^{4}\right) = \log(6561)\]

     When the argument of a log function has an exponent, that exponent can be brought down as a factor outside the log function.

     \[4\cdot\log(9) = \log(6561)\]

     We can divide both sides by \(\log(9)\). \[4 = \frac{\log(6561)}{\log(9)}\]

     Apply the change of base rule. \[4 = \log_{9}(6561)\] Switch the sides.

     \[\log_{9}(6561) = 4\]

     So, \(a=9\) and \(b=6561\) and \(c=4\).

     
     
108. 
     

     Question

     prevnext

     You can verify the exponential relationship below. \[4^{6} = 4096\]

     These same three numbers can be used to satisfy the following logarithmic equation.

     \[\log_{a}(b) = c\]

     Determine values for \(a\), \(b\), and \(c\).

     • \(a =\) -------- 464096
     • \(b =\) -------- 464096
     • \(c =\) -------- 464096
     
     
     
     

     Solution

     Show solution

     Personally, I prefer thinking this through algebraically. We can take a log of both sides.

     \[\log\left(4^{6}\right) = \log(4096)\]

     When the argument of a log function has an exponent, that exponent can be brought down as a factor outside the log function.

     \[6\cdot\log(4) = \log(4096)\]

     We can divide both sides by \(\log(4)\). \[6 = \frac{\log(4096)}{\log(4)}\]

     Apply the change of base rule. \[6 = \log_{4}(4096)\] Switch the sides.

     \[\log_{4}(4096) = 6\]

     So, \(a=4\) and \(b=4096\) and \(c=6\).

     
     
109. 
     

     Question

     prevnext

     You can verify the exponential relationship below. \[10^{2} = 100\]

     These same three numbers can be used to satisfy the following logarithmic equation.

     \[\log_{a}(b) = c\]

     Determine values for \(a\), \(b\), and \(c\).

     • \(a =\) -------- 210100
     • \(b =\) -------- 210100
     • \(c =\) -------- 210100
     
     
     
     

     Solution

     Show solution

     Personally, I prefer thinking this through algebraically. We can take a log of both sides.

     \[\log\left(10^{2}\right) = \log(100)\]

     When the argument of a log function has an exponent, that exponent can be brought down as a factor outside the log function.

     \[2\cdot\log(10) = \log(100)\]

     We can divide both sides by \(\log(10)\). \[2 = \frac{\log(100)}{\log(10)}\]

     Apply the change of base rule. \[2 = \log_{10}(100)\] Switch the sides.

     \[\log_{10}(100) = 2\]

     So, \(a=10\) and \(b=100\) and \(c=2\).

     
     
110. 
     

     Question

     prevnext

     You can verify the exponential relationship below. \[4^{5} = 1024\]

     These same three numbers can be used to satisfy the following logarithmic equation.

     \[\log_{a}(b) = c\]

     Determine values for \(a\), \(b\), and \(c\).

     • \(a =\) -------- 451024
     • \(b =\) -------- 451024
     • \(c =\) -------- 451024
     
     
     
     

     Solution

     Show solution

     Personally, I prefer thinking this through algebraically. We can take a log of both sides.

     \[\log\left(4^{5}\right) = \log(1024)\]

     When the argument of a log function has an exponent, that exponent can be brought down as a factor outside the log function.

     \[5\cdot\log(4) = \log(1024)\]

     We can divide both sides by \(\log(4)\). \[5 = \frac{\log(1024)}{\log(4)}\]

     Apply the change of base rule. \[5 = \log_{4}(1024)\] Switch the sides.

     \[\log_{4}(1024) = 5\]

     So, \(a=4\) and \(b=1024\) and \(c=5\).

     
     
111. 
     

     Question

     prevnext

     In a school, there are 84 students. Each student can be in bike club, crochet club, both, or neither. The amounts are represented in the Venn diagram below.

     
     

     Let \(B\) represent the event that a randomly selected student is in bike club, and let \(C\) represent the event that a random student is in crochet club.

     • \(P(C)=\) -------- 0.3809523809523810.5423728813559320.5714285714285710.6666666666666670.7023809523809520.892857142857143

     • \(P(B|C)=\) -------- 0.3809523809523810.5423728813559320.5714285714285710.6666666666666670.7023809523809520.892857142857143

     • \(P(B)=\) -------- 0.3809523809523810.5423728813559320.5714285714285710.6666666666666670.7023809523809520.892857142857143

     • \(P(B\cap C)=\) -------- 0.3809523809523810.5423728813559320.5714285714285710.6666666666666670.7023809523809520.892857142857143

     • \(P(B\cup C)=\) -------- 0.3809523809523810.5423728813559320.5714285714285710.6666666666666670.7023809523809520.892857142857143

     • \(P(C|B)=\) -------- 0.3809523809523810.5423728813559320.5714285714285710.6666666666666670.7023809523809520.892857142857143

     • Are events \(B\) and \(C\) independent? -------- Yes, B and C are independent.No, B and C are not independent.

     
     
     
     

     Solution

     Show solution

     The three symbols (and related concepts) we are learning are:

     • Intersection, and, \(\cap\)
     • Union, or, \(\cup\)
     • Conditional, given, \(|\)

     To calculate the probabilities (in a more organized order):

     • \(P(B) = \frac{16+32}{84} = 0.5714286\)
     • \(P(C) = \frac{27+32}{84} = 0.702381\)
     • \(P(B\cap C) = \frac{32}{84} = 0.3809524\)
     • \(P(B\cup C) = \frac{16+32+27}{84} = 0.8928571\)
     • \(P(B|C) = \frac{32}{32+27} = 0.5423729\)
     • \(P(C|B) = \frac{32}{32+16} = 0.6666667\)

     The events are independent if and only if \(P(B|C)~=~P(B)\). So, no, they are not independent.

     
     
112. 
     

     Question

     prevnext

     In a school, there are 131 students. Each student can be in bike club, crochet club, both, or neither. The amounts are represented in the Venn diagram below.

     
     

     Let \(B\) represent the event that a randomly selected student is in bike club, and let \(C\) represent the event that a random student is in crochet club.

     • \(P(B|C)=\) -------- 0.2595419847328240.3206106870229010.4594594594594590.5648854961832060.6259541984732820.80952380952381

     • \(P(B\cup C)=\) -------- 0.2595419847328240.3206106870229010.4594594594594590.5648854961832060.6259541984732820.80952380952381

     • \(P(C|B)=\) -------- 0.2595419847328240.3206106870229010.4594594594594590.5648854961832060.6259541984732820.80952380952381

     • \(P(B)=\) -------- 0.2595419847328240.3206106870229010.4594594594594590.5648854961832060.6259541984732820.80952380952381

     • \(P(C)=\) -------- 0.2595419847328240.3206106870229010.4594594594594590.5648854961832060.6259541984732820.80952380952381

     • \(P(B\cap C)=\) -------- 0.2595419847328240.3206106870229010.4594594594594590.5648854961832060.6259541984732820.80952380952381

     • Are events \(B\) and \(C\) independent? -------- Yes, B and C are independent.No, B and C are not independent.

     
     
     
     

     Solution

     Show solution

     The three symbols (and related concepts) we are learning are:

     • Intersection, and, \(\cap\)
     • Union, or, \(\cup\)
     • Conditional, given, \(|\)

     To calculate the probabilities (in a more organized order):

     • \(P(B) = \frac{8+34}{131} = 0.3206107\)
     • \(P(C) = \frac{40+34}{131} = 0.5648855\)
     • \(P(B\cap C) = \frac{34}{131} = 0.259542\)
     • \(P(B\cup C) = \frac{8+34+40}{131} = 0.6259542\)
     • \(P(B|C) = \frac{34}{34+40} = 0.4594595\)
     • \(P(C|B) = \frac{34}{34+8} = 0.8095238\)

     The events are independent if and only if \(P(B|C)~=~P(B)\). So, no, they are not independent.

     
     
113. 
     

     Question

     prevnext

     In a school, there are 129 students. Each student can be in bike club, crochet club, both, or neither. The amounts are represented in the Venn diagram below.

     
     

     Let \(B\) represent the event that a randomly selected student is in bike club, and let \(C\) represent the event that a random student is in crochet club.

     • \(P(B\cup C)=\) -------- 0.1860465116279070.2945736434108530.3333333333333330.3720930232558140.5581395348837210.705426356589147

     • \(P(C)=\) -------- 0.1860465116279070.2945736434108530.3333333333333330.3720930232558140.5581395348837210.705426356589147

     • \(P(B\cap C)=\) -------- 0.1860465116279070.2945736434108530.3333333333333330.3720930232558140.5581395348837210.705426356589147

     • \(P(B)=\) -------- 0.1860465116279070.2945736434108530.3333333333333330.3720930232558140.5581395348837210.705426356589147

     • \(P(B|C)=\) -------- 0.1860465116279070.2945736434108530.3333333333333330.3720930232558140.5581395348837210.705426356589147

     • \(P(C|B)=\) -------- 0.1860465116279070.2945736434108530.3333333333333330.3720930232558140.5581395348837210.705426356589147

     • Are events \(B\) and \(C\) independent? -------- Yes, B and C are independent.No, B and C are not independent.

     
     
     
     

     Solution

     Show solution

     The three symbols (and related concepts) we are learning are:

     • Intersection, and, \(\cap\)
     • Union, or, \(\cup\)
     • Conditional, given, \(|\)

     To calculate the probabilities (in a more organized order):

     • \(P(B) = \frac{19+24}{129} = 0.3333333\)
     • \(P(C) = \frac{48+24}{129} = 0.5581395\)
     • \(P(B\cap C) = \frac{24}{129} = 0.1860465\)
     • \(P(B\cup C) = \frac{19+24+48}{129} = 0.7054264\)
     • \(P(B|C) = \frac{24}{24+48} = 0.3333333\)
     • \(P(C|B) = \frac{24}{24+19} = 0.5581395\)

     The events are independent if and only if \(P(B|C)~=~P(B)\). So, yes, they are independent.

     
     
114. 
     

     Question

     prevnext

     In a school, there are 66 students. Each student can be in bike club, crochet club, both, or neither. The amounts are represented in the Venn diagram below.

     
     

     Let \(B\) represent the event that a randomly selected student is in bike club, and let \(C\) represent the event that a random student is in crochet club.

     • \(P(B\cup C)=\) -------- 0.2121212121212120.2424242424242420.2857142857142860.3636363636363640.6666666666666670.787878787878788

     • \(P(B|C)=\) -------- 0.2121212121212120.2424242424242420.2857142857142860.3636363636363640.6666666666666670.787878787878788

     • \(P(C|B)=\) -------- 0.2121212121212120.2424242424242420.2857142857142860.3636363636363640.6666666666666670.787878787878788

     • \(P(C)=\) -------- 0.2121212121212120.2424242424242420.2857142857142860.3636363636363640.6666666666666670.787878787878788

     • \(P(B)=\) -------- 0.2121212121212120.2424242424242420.2857142857142860.3636363636363640.6666666666666670.787878787878788

     • \(P(B\cap C)=\) -------- 0.2121212121212120.2424242424242420.2857142857142860.3636363636363640.6666666666666670.787878787878788

     • Are events \(B\) and \(C\) independent? -------- Yes, B and C are independent.No, B and C are not independent.

     
     
     
     

     Solution

     Show solution

     The three symbols (and related concepts) we are learning are:

     • Intersection, and, \(\cap\)
     • Union, or, \(\cup\)
     • Conditional, given, \(|\)

     To calculate the probabilities (in a more organized order):

     • \(P(B) = \frac{8+16}{66} = 0.3636364\)
     • \(P(C) = \frac{28+16}{66} = 0.6666667\)
     • \(P(B\cap C) = \frac{16}{66} = 0.2424242\)
     • \(P(B\cup C) = \frac{8+16+28}{66} = 0.7878788\)
     • \(P(B|C) = \frac{16}{16+28} = 0.3636364\)
     • \(P(C|B) = \frac{16}{16+8} = 0.6666667\)

     The events are independent if and only if \(P(B|C)~=~P(B)\). So, yes, they are independent.

     
     
115. 
     

     Question

     prevnext

     In a school, there are 114 students. Each student can be in bike club, crochet club, both, or neither. The amounts are represented in the Venn diagram below.

     
     

     Let \(B\) represent the event that a randomly selected student is in bike club, and let \(C\) represent the event that a random student is in crochet club.

     • \(P(C|B)=\) -------- 0.438596491228070.4824561403508770.5494505494505490.7982456140350880.8421052631578950.909090909090909

     • \(P(B\cup C)=\) -------- 0.438596491228070.4824561403508770.5494505494505490.7982456140350880.8421052631578950.909090909090909

     • \(P(B|C)=\) -------- 0.438596491228070.4824561403508770.5494505494505490.7982456140350880.8421052631578950.909090909090909

     • \(P(B)=\) -------- 0.438596491228070.4824561403508770.5494505494505490.7982456140350880.8421052631578950.909090909090909

     • \(P(B\cap C)=\) -------- 0.438596491228070.4824561403508770.5494505494505490.7982456140350880.8421052631578950.909090909090909

     • \(P(C)=\) -------- 0.438596491228070.4824561403508770.5494505494505490.7982456140350880.8421052631578950.909090909090909

     • Are events \(B\) and \(C\) independent? -------- Yes, B and C are independent.No, B and C are not independent.

     
     
     
     

     Solution

     Show solution

     The three symbols (and related concepts) we are learning are:

     • Intersection, and, \(\cap\)
     • Union, or, \(\cup\)
     • Conditional, given, \(|\)

     To calculate the probabilities (in a more organized order):

     • \(P(B) = \frac{41+50}{114} = 0.7982456\)
     • \(P(C) = \frac{5+50}{114} = 0.4824561\)
     • \(P(B\cap C) = \frac{50}{114} = 0.4385965\)
     • \(P(B\cup C) = \frac{41+50+5}{114} = 0.8421053\)
     • \(P(B|C) = \frac{50}{50+5} = 0.9090909\)
     • \(P(C|B) = \frac{50}{50+41} = 0.5494505\)

     The events are independent if and only if \(P(B|C)~=~P(B)\). So, no, they are not independent.

     
     
116. 
     

     Question

     prevnext

     In a school, there are 72 students. Each student can be in bike club, crochet club, both, or neither. The amounts are represented in the Venn diagram below.

     
     

     Let \(B\) represent the event that a randomly selected student is in bike club, and let \(C\) represent the event that a random student is in crochet club.

     • \(P(B|C)=\) -------- 0.04166666666666670.06944444444444440.1111111111111110.3333333333333330.3750.444444444444444

     • \(P(B)=\) -------- 0.04166666666666670.06944444444444440.1111111111111110.3333333333333330.3750.444444444444444

     • \(P(B\cup C)=\) -------- 0.04166666666666670.06944444444444440.1111111111111110.3333333333333330.3750.444444444444444

     • \(P(C)=\) -------- 0.04166666666666670.06944444444444440.1111111111111110.3333333333333330.3750.444444444444444

     • \(P(C|B)=\) -------- 0.04166666666666670.06944444444444440.1111111111111110.3333333333333330.3750.444444444444444

     • \(P(B\cap C)=\) -------- 0.04166666666666670.06944444444444440.1111111111111110.3333333333333330.3750.444444444444444

     • Are events \(B\) and \(C\) independent? -------- Yes, B and C are independent.No, B and C are not independent.

     
     
     
     

     Solution

     Show solution

     The three symbols (and related concepts) we are learning are:

     • Intersection, and, \(\cap\)
     • Union, or, \(\cup\)
     • Conditional, given, \(|\)

     To calculate the probabilities (in a more organized order):

     • \(P(B) = \frac{24+3}{72} = 0.375\)
     • \(P(C) = \frac{5+3}{72} = 0.1111111\)
     • \(P(B\cap C) = \frac{3}{72} = 0.0416667\)
     • \(P(B\cup C) = \frac{24+3+5}{72} = 0.4444444\)
     • \(P(B|C) = \frac{3}{3+5} = 0.375\)
     • \(P(C|B) = \frac{3}{3+24} = 0.1111111\)

     The events are independent if and only if \(P(B|C)~=~P(B)\). So, yes, they are independent.

     
     
117. 
     

     Question

     prevnext

     In a school, there are 78 students. Each student can be in bike club, crochet club, both, or neither. The amounts are represented in the Venn diagram below.

     
     

     Let \(B\) represent the event that a randomly selected student is in bike club, and let \(C\) represent the event that a random student is in crochet club.

     • \(P(B\cup C)=\) -------- 0.2051282051282050.2307692307692310.4615384615384620.6666666666666670.6923076923076920.897435897435897

     • \(P(B)=\) -------- 0.2051282051282050.2307692307692310.4615384615384620.6666666666666670.6923076923076920.897435897435897

     • \(P(C|B)=\) -------- 0.2051282051282050.2307692307692310.4615384615384620.6666666666666670.6923076923076920.897435897435897

     • \(P(C)=\) -------- 0.2051282051282050.2307692307692310.4615384615384620.6666666666666670.6923076923076920.897435897435897

     • \(P(B|C)=\) -------- 0.2051282051282050.2307692307692310.4615384615384620.6666666666666670.6923076923076920.897435897435897

     • \(P(B\cap C)=\) -------- 0.2051282051282050.2307692307692310.4615384615384620.6666666666666670.6923076923076920.897435897435897

     • Are events \(B\) and \(C\) independent? -------- Yes, B and C are independent.No, B and C are not independent.

     
     
     
     

     Solution

     Show solution

     The three symbols (and related concepts) we are learning are:

     • Intersection, and, \(\cap\)
     • Union, or, \(\cup\)
     • Conditional, given, \(|\)

     To calculate the probabilities (in a more organized order):

     • \(P(B) = \frac{18+36}{78} = 0.6923077\)
     • \(P(C) = \frac{16+36}{78} = 0.6666667\)
     • \(P(B\cap C) = \frac{36}{78} = 0.4615385\)
     • \(P(B\cup C) = \frac{18+36+16}{78} = 0.8974359\)
     • \(P(B|C) = \frac{36}{36+16} = 0.6923077\)
     • \(P(C|B) = \frac{36}{36+18} = 0.6666667\)

     The events are independent if and only if \(P(B|C)~=~P(B)\). So, yes, they are independent.

     
     
118. 
     

     Question

     prevnext

     In a school, there are 100 students. Each student can be in bike club, crochet club, both, or neither. The amounts are represented in the Venn diagram below.

     
     

     Let \(B\) represent the event that a randomly selected student is in bike club, and let \(C\) represent the event that a random student is in crochet club.

     • \(P(B|C)=\) -------- 0.140.330.420.560.750.89

     • \(P(C|B)=\) -------- 0.140.330.420.560.750.89

     • \(P(C)=\) -------- 0.140.330.420.560.750.89

     • \(P(B\cup C)=\) -------- 0.140.330.420.560.750.89

     • \(P(B\cap C)=\) -------- 0.140.330.420.560.750.89

     • \(P(B)=\) -------- 0.140.330.420.560.750.89

     • Are events \(B\) and \(C\) independent? -------- Yes, B and C are independent.No, B and C are not independent.

     
     
     
     

     Solution

     Show solution

     The three symbols (and related concepts) we are learning are:

     • Intersection, and, \(\cap\)
     • Union, or, \(\cup\)
     • Conditional, given, \(|\)

     To calculate the probabilities (in a more organized order):

     • \(P(B) = \frac{14+42}{100} = 0.56\)
     • \(P(C) = \frac{33+42}{100} = 0.75\)
     • \(P(B\cap C) = \frac{42}{100} = 0.42\)
     • \(P(B\cup C) = \frac{14+42+33}{100} = 0.89\)
     • \(P(B|C) = \frac{42}{42+33} = 0.56\)
     • \(P(C|B) = \frac{42}{42+14} = 0.75\)

     The events are independent if and only if \(P(B|C)~=~P(B)\). So, yes, they are independent.

     
     
119. 
     

     Question

     prevnext

     In a school, there are 77 students. Each student can be in bike club, crochet club, both, or neither. The amounts are represented in the Venn diagram below.

     
     

     Let \(B\) represent the event that a randomly selected student is in bike club, and let \(C\) represent the event that a random student is in crochet club.

     • \(P(B\cup C)=\) -------- 0.06493506493506490.1428571428571430.20.389610389610390.4545454545454550.532467532467532

     • \(P(C|B)=\) -------- 0.06493506493506490.1428571428571430.20.389610389610390.4545454545454550.532467532467532

     • \(P(B)=\) -------- 0.06493506493506490.1428571428571430.20.389610389610390.4545454545454550.532467532467532

     • \(P(B\cap C)=\) -------- 0.06493506493506490.1428571428571430.20.389610389610390.4545454545454550.532467532467532

     • \(P(C)=\) -------- 0.06493506493506490.1428571428571430.20.389610389610390.4545454545454550.532467532467532

     • \(P(B|C)=\) -------- 0.06493506493506490.1428571428571430.20.389610389610390.4545454545454550.532467532467532

     • Are events \(B\) and \(C\) independent? -------- Yes, B and C are independent.No, B and C are not independent.

     
     
     
     

     Solution

     Show solution

     The three symbols (and related concepts) we are learning are:

     • Intersection, and, \(\cap\)
     • Union, or, \(\cup\)
     • Conditional, given, \(|\)

     To calculate the probabilities (in a more organized order):

     • \(P(B) = \frac{30+5}{77} = 0.4545455\)
     • \(P(C) = \frac{6+5}{77} = 0.1428571\)
     • \(P(B\cap C) = \frac{5}{77} = 0.0649351\)
     • \(P(B\cup C) = \frac{30+5+6}{77} = 0.5324675\)
     • \(P(B|C) = \frac{5}{5+6} = 0.4545455\)
     • \(P(C|B) = \frac{5}{5+30} = 0.1428571\)

     The events are independent if and only if \(P(B|C)~=~P(B)\). So, yes, they are independent.

     
     
120. 
     

     Question

     prevnext

     In a school, there are 53 students. Each student can be in bike club, crochet club, both, or neither. The amounts are represented in the Venn diagram below.

     
     

     Let \(B\) represent the event that a randomly selected student is in bike club, and let \(C\) represent the event that a random student is in crochet club.

     • \(P(B\cup C)=\) -------- 0.07547169811320750.1818181818181820.2641509433962260.2857142857142860.4150943396226420.60377358490566

     • \(P(B\cap C)=\) -------- 0.07547169811320750.1818181818181820.2641509433962260.2857142857142860.4150943396226420.60377358490566

     • \(P(C|B)=\) -------- 0.07547169811320750.1818181818181820.2641509433962260.2857142857142860.4150943396226420.60377358490566

     • \(P(B)=\) -------- 0.07547169811320750.1818181818181820.2641509433962260.2857142857142860.4150943396226420.60377358490566

     • \(P(B|C)=\) -------- 0.07547169811320750.1818181818181820.2641509433962260.2857142857142860.4150943396226420.60377358490566

     • \(P(C)=\) -------- 0.07547169811320750.1818181818181820.2641509433962260.2857142857142860.4150943396226420.60377358490566

     • Are events \(B\) and \(C\) independent? -------- Yes, B and C are independent.No, B and C are not independent.

     
     
     
     

     Solution

     Show solution

     The three symbols (and related concepts) we are learning are:

     • Intersection, and, \(\cap\)
     • Union, or, \(\cup\)
     • Conditional, given, \(|\)

     To calculate the probabilities (in a more organized order):

     • \(P(B) = \frac{10+4}{53} = 0.2641509\)
     • \(P(C) = \frac{18+4}{53} = 0.4150943\)
     • \(P(B\cap C) = \frac{4}{53} = 0.0754717\)
     • \(P(B\cup C) = \frac{10+4+18}{53} = 0.6037736\)
     • \(P(B|C) = \frac{4}{4+18} = 0.1818182\)
     • \(P(C|B) = \frac{4}{4+10} = 0.2857143\)

     The events are independent if and only if \(P(B|C)~=~P(B)\). So, no, they are not independent.

     
     
121. 
     

     Question

     prevnext

     The inner dimensions of a frame are 40 inches by 28 inches. An artist hopes to print an image with an aspect ratio of 2.05 such that a uniformly wide mat fits around the image.

     
     

     Find \(b\), the width of the mat in inches. The tolerance is \(\pm 0.01\) inches.

     
     

     Solution

     Show solution

     A system of equations can be written from the frame’s width and height:

     \[40 = 2.05y+2b\] \[28 = y+2b\] Subtract the two equations.

     \[12 = 1.05y\]

     Divide both sides by 1.05.

     \[11.4285714 = y\]

     To find \(b\), you can rearrange the second equation from the system, and plug in the value of \(y\). \[b ~=~ \frac{28-y}{2} ~=~ \frac{28-11.4285714}{2} ~=~ 8.2857143\]

     
     
122. 
     

     Question

     prevnext

     The inner dimensions of a frame are 32 inches by 18 inches. An artist hopes to print an image with an aspect ratio of 2.58 such that a uniformly wide mat fits around the image.

     
     

     Find \(b\), the width of the mat in inches. The tolerance is \(\pm 0.01\) inches.

     
     

     Solution

     Show solution

     A system of equations can be written from the frame’s width and height:

     \[32 = 2.58y+2b\] \[18 = y+2b\] Subtract the two equations.

     \[14 = 1.58y\]

     Divide both sides by 1.58.

     \[8.8607595 = y\]

     To find \(b\), you can rearrange the second equation from the system, and plug in the value of \(y\). \[b ~=~ \frac{18-y}{2} ~=~ \frac{18-8.8607595}{2} ~=~ 4.5696203\]

     
     
123. 
     

     Question

     prevnext

     The inner dimensions of a frame are 25 inches by 18 inches. An artist hopes to print an image with an aspect ratio of 1.75 such that a uniformly wide mat fits around the image.

     
     

     Find \(b\), the width of the mat in inches. The tolerance is \(\pm 0.01\) inches.

     
     

     Solution

     Show solution

     A system of equations can be written from the frame’s width and height:

     \[25 = 1.75y+2b\] \[18 = y+2b\] Subtract the two equations.

     \[7 = 0.75y\]

     Divide both sides by 0.75.

     \[9.3333333 = y\]

     To find \(b\), you can rearrange the second equation from the system, and plug in the value of \(y\). \[b ~=~ \frac{18-y}{2} ~=~ \frac{18-9.3333333}{2} ~=~ 4.3333333\]

     
     
124. 
     

     Question

     prevnext

     The inner dimensions of a frame are 36 inches by 21 inches. An artist hopes to print an image with an aspect ratio of 2.24 such that a uniformly wide mat fits around the image.

     
     

     Find \(b\), the width of the mat in inches. The tolerance is \(\pm 0.01\) inches.

     
     

     Solution

     Show solution

     A system of equations can be written from the frame’s width and height:

     \[36 = 2.24y+2b\] \[21 = y+2b\] Subtract the two equations.

     \[15 = 1.24y\]

     Divide both sides by 1.24.

     \[12.0967742 = y\]

     To find \(b\), you can rearrange the second equation from the system, and plug in the value of \(y\). \[b ~=~ \frac{21-y}{2} ~=~ \frac{21-12.0967742}{2} ~=~ 4.4516129\]

     
     
125. 
     

     Question

     prevnext

     The inner dimensions of a frame are 19 inches by 14 inches. An artist hopes to print an image with an aspect ratio of 1.65 such that a uniformly wide mat fits around the image.

     
     

     Find \(b\), the width of the mat in inches. The tolerance is \(\pm 0.01\) inches.

     
     

     Solution

     Show solution

     A system of equations can be written from the frame’s width and height:

     \[19 = 1.65y+2b\] \[14 = y+2b\] Subtract the two equations.

     \[5 = 0.65y\]

     Divide both sides by 0.65.

     \[7.6923077 = y\]

     To find \(b\), you can rearrange the second equation from the system, and plug in the value of \(y\). \[b ~=~ \frac{14-y}{2} ~=~ \frac{14-7.6923077}{2} ~=~ 3.1538462\]

     
     
126. 
     

     Question

     prevnext

     The inner dimensions of a frame are 34 inches by 28 inches. An artist hopes to print an image with an aspect ratio of 1.26 such that a uniformly wide mat fits around the image.

     
     

     Find \(y\), the height of the image in inches. The tolerance is \(\pm 0.01\) inches.

     
     

     Solution

     Show solution

     A system of equations can be written from the frame’s width and height:

     \[34 = 1.26y+2b\] \[28 = y+2b\] Subtract the two equations.

     \[6 = 0.26y\]

     Divide both sides by 0.26.

     \[23.0769231 = y\]

     To find \(b\), you can rearrange the second equation from the system, and plug in the value of \(y\). \[b ~=~ \frac{28-y}{2} ~=~ \frac{28-23.0769231}{2} ~=~ 2.4615385\]

     
     
127. 
     

     Question

     prevnext

     The inner dimensions of a frame are 33 inches by 14 inches. An artist hopes to print an image with an aspect ratio of 3.2 such that a uniformly wide mat fits around the image.

     
     

     Find \(y\), the height of the image in inches. The tolerance is \(\pm 0.01\) inches.

     
     

     Solution

     Show solution

     A system of equations can be written from the frame’s width and height:

     \[33 = 3.2y+2b\] \[14 = y+2b\] Subtract the two equations.

     \[19 = 2.2y\]

     Divide both sides by 2.2.

     \[8.6363636 = y\]

     To find \(b\), you can rearrange the second equation from the system, and plug in the value of \(y\). \[b ~=~ \frac{14-y}{2} ~=~ \frac{14-8.6363636}{2} ~=~ 2.6818182\]

     
     
128. 
     

     Question

     prevnext

     The inner dimensions of a frame are 40 inches by 20 inches. An artist hopes to print an image with an aspect ratio of 2.28 such that a uniformly wide mat fits around the image.

     
     

     Find \(b\), the width of the mat in inches. The tolerance is \(\pm 0.01\) inches.

     
     

     Solution

     Show solution

     A system of equations can be written from the frame’s width and height:

     \[40 = 2.28y+2b\] \[20 = y+2b\] Subtract the two equations.

     \[20 = 1.28y\]

     Divide both sides by 1.28.

     \[15.625 = y\]

     To find \(b\), you can rearrange the second equation from the system, and plug in the value of \(y\). \[b ~=~ \frac{20-y}{2} ~=~ \frac{20-15.625}{2} ~=~ 2.1875\]

     
     
129. 
     

     Question

     prevnext

     The inner dimensions of a frame are 31 inches by 21 inches. An artist hopes to print an image with an aspect ratio of 1.59 such that a uniformly wide mat fits around the image.

     
     

     Find \(b\), the width of the mat in inches. The tolerance is \(\pm 0.01\) inches.

     
     

     Solution

     Show solution

     A system of equations can be written from the frame’s width and height:

     \[31 = 1.59y+2b\] \[21 = y+2b\] Subtract the two equations.

     \[10 = 0.59y\]

     Divide both sides by 0.59.

     \[16.9491525 = y\]

     To find \(b\), you can rearrange the second equation from the system, and plug in the value of \(y\). \[b ~=~ \frac{21-y}{2} ~=~ \frac{21-16.9491525}{2} ~=~ 2.0254237\]

     
     
130. 
     

     Question

     prevnext

     The inner dimensions of a frame are 22 inches by 17 inches. An artist hopes to print an image with an aspect ratio of 1.58 such that a uniformly wide mat fits around the image.

     
     

     Find \(y\), the height of the image in inches. The tolerance is \(\pm 0.01\) inches.

     
     

     Solution

     Show solution

     A system of equations can be written from the frame’s width and height:

     \[22 = 1.58y+2b\] \[17 = y+2b\] Subtract the two equations.

     \[5 = 0.58y\]

     Divide both sides by 0.58.

     \[8.6206897 = y\]

     To find \(b\), you can rearrange the second equation from the system, and plug in the value of \(y\). \[b ~=~ \frac{17-y}{2} ~=~ \frac{17-8.6206897}{2} ~=~ 4.1896552\]